Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis...
Transcript of Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis...
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Geometry of R Roots of 9×9 PolynomialSystems
Luis FelicianoTexas A&M University
July 16, 2018REU: DMS – 1757872
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 2: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/2.jpg)
Motivation
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10
A Quadratic Pentanomial 2x2 system!
Solving polynomial equations becomes more and morecomplicated as we increase the number of terms and variables.
Today we’ll take a look at some constructions that give usrough approximations for roots in a fraction of the time!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 3: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/3.jpg)
Motivation
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10
A Quadratic Pentanomial 2x2 system!
Solving polynomial equations becomes more and morecomplicated as we increase the number of terms and variables.
Today we’ll take a look at some constructions that give usrough approximations for roots in a fraction of the time!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 4: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/4.jpg)
Motivation
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10
A Quadratic Pentanomial 2x2 system!
Solving polynomial equations becomes more and morecomplicated as we increase the number of terms and variables.
Today we’ll take a look at some constructions that give usrough approximations for roots in a fraction of the time!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 5: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/5.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segment PQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 6: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/6.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segment PQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 7: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/7.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 8: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/8.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 9: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/9.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 10: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/10.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 11: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/11.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 12: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/12.jpg)
A Quick Review
A convex set is a set of points such that, given any two pointsP ,Q , then the line segmentPQ is also in the set
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 13: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/13.jpg)
Convex Hull
Denoted by conv{·}
For any S ⊂ Rn
conv{S} := smallest convex set containing S
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 14: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/14.jpg)
Convex Hull
Denoted by conv{·}
For any S ⊂ Rn
conv{S} := smallest convex set containing S
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 15: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/15.jpg)
Convex Hull
Denoted by conv{·}
For any S ⊂ Rn
conv{S} := smallest convex set containing S
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 16: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/16.jpg)
Convex Hull
Denoted by conv{·}
For any S ⊂ Rn
conv{S} := smallest convex set containing S
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 17: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/17.jpg)
Convex Hull
Denoted by conv{·}
For any S ⊂ Rn
conv{S} := smallest convex set containing S
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 18: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/18.jpg)
Convex Hull
Denoted by conv{·}
For any S ⊂ Rn
conv{S} := smallest convex set containing S
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 19: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/19.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 20: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/20.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 21: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/21.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 22: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/22.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 23: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/23.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 24: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/24.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 25: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/25.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 26: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/26.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 27: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/27.jpg)
Newton Polytope
Denoted by Newt(·)
f(x) =t∑
i=1cix
ai
where xai = xa1,i1 x
a2,i2 · · · xan,i
n
Newt(f) := conv{ai | ci 6= 0}
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 28: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/28.jpg)
Newton Polytope
Although we didn’t make use of the following in our research...
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 29: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/29.jpg)
Newton Polytope
Although we didn’t make use of the following in our research...
The volume of the Newton polytope can be used to computethe degree of the corresponding hypersurface, and via mixedvolumes, the number of roots of systems of equations!
Ex:f(x, y) = 1 + x2 + y3 − 100xy
= 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 30: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/30.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 31: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/31.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 32: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/32.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 33: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/33.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 34: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/34.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1))
, (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 35: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/35.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 36: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/36.jpg)
Archimedean Newton Polytope
Denoted by ArchNewt(f)
ArchNewt(f) := conv{(ai,−Log|ci|) | i ∈ {1, . . . , t}, ci 6= 0}
f(x, y) = 1 ∗ x0 ∗ y0 + 1 ∗ x2 ∗ y0 + 1 ∗ x0 ∗ y3 − 100 ∗ x1 ∗ y1
⇒ ArchNewt(f) =
conv{(0, 0,−Log(1)), (2, 0,−Log(1)), (0, 3,−Log(1)), (1, 1,−Log(100))}
⇒ conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 37: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/37.jpg)
conv{(0, 0, 0), (2, 0, 0), (0, 3, 0), (1, 1,−Log(100))}
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 38: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/38.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 39: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/39.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 40: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/40.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 41: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/41.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 42: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/42.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 43: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/43.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 44: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/44.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 45: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/45.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 46: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/46.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 47: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/47.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 48: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/48.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
This gives us a triangulation of our Newton Polytope!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 49: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/49.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
This gives us a triangulation of our Newton Polytope!
We take the outer normals of these lower faces
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 50: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/50.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
This gives us a triangulation of our Newton Polytope!
We take the outer normals of these lower faces
→We normalize them to be of the form (w,−1), and take
w to be a vertex of ArchTrop(f)!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 51: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/51.jpg)
Archimedean Tropical Variety
Let’s project the lower faces of ArchNewt(f) onto the xy-plane
This gives us a triangulation of our Newton Polytope!
We take the outer normals of these lower faces
→We normalize them to be of the form (w,−1), and take
w to be a vertex of ArchTrop(f)!
→ Roughly* translates to a point in each triangle!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 52: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/52.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 53: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/53.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
→ The outer normals of the edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 54: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/54.jpg)
Archimedean Tropical Variety
We need two things to construct ArchTrop(f)
→ The outer normals of ArchNewt(f) that point downwards
→ The outer normals of the exterior edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 55: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/55.jpg)
Archimedean Tropical Variety
Putting these two together, we get...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 56: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/56.jpg)
Archimedean Tropical Variety
Putting these two together, we get...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 57: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/57.jpg)
Archimedean Tropical Variety - Roughly*
The vertices of ArchTrop(f) are dual to the triangulation ofNewt(f) induced by the lower faces of ArchNewt(f)
The rays are dual to the edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 58: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/58.jpg)
Archimedean Tropical Variety - Roughly*
The vertices of ArchTrop(f) are dual to the triangulation ofNewt(f) induced by the lower faces of ArchNewt(f)
The rays are dual to the edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 59: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/59.jpg)
Archimedean Tropical Variety - Roughly*
The vertices of ArchTrop(f) are dual to the triangulation ofNewt(f) induced by the lower faces of ArchNewt(f)
The rays are dual to the edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 60: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/60.jpg)
Archimedean Tropical Variety - Roughly*
The vertices of ArchTrop(f) are dual to the triangulation ofNewt(f) induced by the lower faces of ArchNewt(f)
The rays are dual to the edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 61: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/61.jpg)
Archimedean Tropical Variety - Roughly*
The vertices of ArchTrop(f) are dual to the triangulation ofNewt(f) induced by the lower faces of ArchNewt(f)
The rays are dual to the edges of Newt(f)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 62: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/62.jpg)
Archimedean Tropical Variety
ArchTrop(f) gives us metric information about the roots andareas where we can find constant isotopy types!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 63: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/63.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 64: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/64.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 65: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/65.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 66: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/66.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 67: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/67.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 68: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/68.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 69: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/69.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 70: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/70.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of roots
ArchTrop(f) can do this for more general curves
Ex:f(x, y) = 1 + x2 + y3 − cxy (c > 0)
⇒ The zero set of f(x, y) is either ∅, a point, or an oval!
⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 71: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/71.jpg)
A Word on Isotopy Types
Much like how the quadratic discriminant b2 − 4ac gives usinformation about the number of rootsArchTrop(f) can do this for more general curvesEx:f(x, y) = 1 + x2 + y3 − cxy (c > 0)⇒ The zero set of f(x, y) is either ∅, a point, or an oval!⇒ This occurs when c < 6
213 3
12
, c = 6
213 3
12
, c > 6
213 3
12
, respectively
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 72: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/72.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 73: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/73.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 74: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/74.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 75: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/75.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 76: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/76.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 77: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/77.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
More specifically, we are interested in alternating signs!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 78: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/78.jpg)
ArchTrop+(f)
ArchTrop+(f) ⊂ ArchTrop(f)
This time we focus on the signs of our coefficients!
f(x, y) = 1 + x2 + y3 − 100xy
More specifically, we are interested in alternating signs!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 79: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/79.jpg)
Archimedean Tropical Variety
We go from this...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 80: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/80.jpg)
Archimedean Tropical Variety
To this!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 81: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/81.jpg)
Archimedean Tropical Variety
ArchTrop+(f) gives us a piecewise linear function thatresembles the set of positive roots
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 82: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/82.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 83: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/83.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 84: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/84.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 85: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/85.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 86: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/86.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 87: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/87.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 88: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/88.jpg)
A Small Discrepancy...
f(x) = 1− 1.1x + x2
ArchTrop+(f) looks like
But if you look at the discriminant⇒ 1.12 − 4 < 0⇒ f has two non-R roots!
On the other hand, if you look at
{c ∈ R+ | connected zero set of (1− cx + x2) 6= ArchTrop+(f)}
= (0, 2)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 89: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/89.jpg)
Our Research
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 90: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/90.jpg)
Our Research - Newton Polytope
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 91: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/91.jpg)
Our Research - Newton Polytope
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 92: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/92.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
f2(x8, x9) = c6x29 + c7x8x9 + c8x8 + c9x9 + c10
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 93: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/93.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 94: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/94.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 95: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/95.jpg)
Our Research - f1
ArchNewt(f1) given these coefficients is...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 96: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/96.jpg)
Our Research - f1
ArchNewt(f1) given these coefficients is...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 97: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/97.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 98: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/98.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 99: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/99.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 6, c2 = −8, c3 = −3, c4 = 2, c5 = 7
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 100: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/100.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 6, c2 = −8, c3 = −3, c4 = 2, c5 = 7
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 101: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/101.jpg)
Our Research - f1
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 6, c2 = −8, c3 = −3, c4 = 2, c5 = 7
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 102: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/102.jpg)
Our Research
No matter the coefficients, these 5 cases encompass all thepossible triangulations of f1!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 103: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/103.jpg)
Our Research
No matter the coefficients, these 5 cases encompass all thepossible triangulations of f1!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 104: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/104.jpg)
Our Research
f1(x8, x9) = c1x28 + c2x8x9 + c3x8 + c4x9 + c5
Suppose c1 = 1, c2 = −10, c3 = −10, c4 = 2, c5 = 1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 105: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/105.jpg)
Our Research
f2(x8, x9) = c6x8x9 + c7x28 + c8x8 + c9x9 + c10
Suppose c6 = 1, c7 = −10, c8 = −10, c9 = 2, c10 = 1
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 106: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/106.jpg)
Our Research
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 107: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/107.jpg)
A Theorem on ArchTrop(f)
ZC(f) := the Complex zero set of f
Theorem
For any pentanomial f in C[x1, . . . , xn], any point of Log|ZC(f)|is within distance log(4) of some point of ArchTrop(f).
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 108: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/108.jpg)
A Theorem on ArchTrop(f)
ZC(f) := the Complex zero set of f
Theorem
For any pentanomial f in C[x1, . . . , xn], any point of Log|ZC(f)|is within distance log(4) of some point of ArchTrop(f).
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 109: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/109.jpg)
A Theorem on ArchTrop(f)
ZC(f) := the Complex zero set of f
Theorem
For any pentanomial f in C[x1, . . . , xn], any point of Log|ZC(f)|is within distance log(4) of some point of ArchTrop(f).
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 110: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/110.jpg)
A Theorem on ArchTrop+(f)
Z+(f) := the positive zero set of f
Theorem
For any pentanomial f in R[x1, . . . , xn], any point of Log|Z+(f)|is within distance log(4) of some point of ArchTrop+(f).
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 111: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/111.jpg)
A Theorem on ArchTrop+(f)
Z+(f) := the positive zero set of f
Theorem
For any pentanomial f in R[x1, . . . , xn], any point of Log|Z+(f)|is within distance log(4) of some point of ArchTrop+(f).
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 112: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/112.jpg)
Using the same coefficients...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 113: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/113.jpg)
Using the same coefficients...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 114: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/114.jpg)
Theorem
If F is a random real 2× 2 quadratic pentanomial system withsupports having Cayley embedding
A =
[2 1 1 0 0 0 1 1 0 00 1 0 1 0 2 1 0 1 0
],
such that the coefficient vector (c1, . . . , c10) has each ci withmean 0, then with probability at least 41%, F has the samenumber of positive roots as the cardinality ofArchTrop(f1) ∩ArchTrop(f2).
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 115: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/115.jpg)
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 116: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/116.jpg)
Successes!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 117: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/117.jpg)
Successes!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 118: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/118.jpg)
Successes!
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 119: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/119.jpg)
Failures...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 120: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/120.jpg)
Failures...crickets...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 121: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/121.jpg)
Failures...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 122: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/122.jpg)
But why?
Some intuition...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 123: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/123.jpg)
But why?
Some intuition...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 124: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/124.jpg)
But why?
Some intuition...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 125: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/125.jpg)
But why?
Some intuition...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 126: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/126.jpg)
But why?
Some intuition...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 127: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/127.jpg)
But why?
Some intuition...
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 128: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/128.jpg)
A Theorem on the Intersections
Theorem
For any 2× 2 polynomial system non-degenerate F withsupports having Cayley embedding A, the number of nonzeroreal roots of F depends only on the completed signedA-discriminant chamber containing F .
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 129: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/129.jpg)
A Theorem on the Intersections
Theorem
For any 2× 2 polynomial system non-degenerate F withsupports having Cayley embedding A, the number of nonzeroreal roots of F depends only on the completed signedA-discriminant chamber containing F .
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 130: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/130.jpg)
That being said...
We can compute the Hausdorff distance betweenArchTrop(f1) ∩ArchTrop(f2) and Log|Z+(f1)| ∩ Log|Z+(f2)|for 1000 random examples to obtain the following:
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 131: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/131.jpg)
That being said...
We can compute the Hausdorff distance betweenArchTrop(f1) ∩ArchTrop(f2) and Log|Z+(f1)| ∩ Log|Z+(f2)|for 1000 random examples to obtain the following:
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 132: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/132.jpg)
That being said...
We can compute the Hausdorff distance betweenArchTrop(f1) ∩ArchTrop(f2) and Log|Z+(f1)| ∩ Log|Z+(f2)|for 1000 random examples to obtain the following:
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 133: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/133.jpg)
Future Research
1. Generalizing our code
2. Finding the conditions under which
h0(Z+(f)) = h0(ArchTrop+(f))
3. Stability and the Jacobian
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 134: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/134.jpg)
Future Research
1. Generalizing our code
2. Finding the conditions under which
h0(Z+(f)) = h0(ArchTrop+(f))
3. Stability and the Jacobian
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 135: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/135.jpg)
Future Research
1. Generalizing our code
2. Finding the conditions under which
h0(Z+(f)) = h0(ArchTrop+(f))
3. Stability and the Jacobian
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION
![Page 136: Geometry of R Roots of 99 Polynomial Systems...Geometry of R Roots of 9 9 Polynomial Systems Luis Feliciano Texas A&M University July 16, 2018 REU: DMS { 1757872 Luis Feliciano Texas](https://reader034.fdocuments.in/reader034/viewer/2022042322/5f0c06e67e708231d433640e/html5/thumbnails/136.jpg)
Future Research
1. Generalizing our code
2. Finding the conditions under which
h0(Z+(f)) = h0(ArchTrop+(f))
3. Stability and the Jacobian
Luis Feliciano Texas A&M University MATH REU FINAL PRESENTATION