Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain...
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Analytic Combinatorics in Several Variables
Robin Pemantle and Mark Wilson
A of A conference, 30 May, 2013
Pemantle Analytic Combinatorics in Several Variables
![Page 2: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/2.jpg)
About the book
Pemantle Analytic Combinatorics in Several Variables
![Page 3: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/3.jpg)
Pemantle Analytic Combinatorics in Several Variables
![Page 4: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/4.jpg)
Dedication
To the memory of Philippe Flajolet,
on whose shoulders stands all of the work herein.
Pemantle Analytic Combinatorics in Several Variables
![Page 5: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/5.jpg)
The book is in four parts
I General introduction and univariate methods
II Some complex analysis and some algebra
III Multivariate asymptotics
IV Appendices
Pemantle Analytic Combinatorics in Several Variables
![Page 6: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/6.jpg)
The book is in four parts
I General introduction and univariate methods
II Some complex analysis and some algebra
III Multivariate asymptotics
IV Appendices
Pemantle Analytic Combinatorics in Several Variables
![Page 7: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/7.jpg)
The book is in four parts
I General introduction and univariate methods
II Some complex analysis and some algebra
III Multivariate asymptotics
IV Appendices
Pemantle Analytic Combinatorics in Several Variables
![Page 8: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/8.jpg)
The book is in four parts
I General introduction and univariate methods
II Some complex analysis and some algebra
III Multivariate asymptotics
IV Appendices
Pemantle Analytic Combinatorics in Several Variables
![Page 9: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/9.jpg)
The book is in four parts
I General introduction and univariate methods
II Some complex analysis and some algebra
III Multivariate asymptotics
IV Appendices
Pemantle Analytic Combinatorics in Several Variables
![Page 10: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/10.jpg)
The Big Question
How painful will this be?
Can I really use these methods without a ridiculousinvestment of time?
Pemantle Analytic Combinatorics in Several Variables
![Page 11: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/11.jpg)
The Big Question
How painful will this be?
Can I really use these methods without a ridiculousinvestment of time?
Pemantle Analytic Combinatorics in Several Variables
![Page 12: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/12.jpg)
The Big Question
How painful will this be?
Can I really use these methods without a ridiculousinvestment of time?
Pemantle Analytic Combinatorics in Several Variables
![Page 13: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/13.jpg)
Pemantle Analytic Combinatorics in Several Variables
![Page 14: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/14.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 15: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/15.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 16: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/16.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 17: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/17.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 18: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/18.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 19: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/19.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 20: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/20.jpg)
Scope of method
Structures with recursive nature
I Analysis of algorithms
I Various families of trees and other graphs
I Probability: random walks, queuing theory, etc.
I Stat mech: particle ensembles, quantum walks, etc.
I Tilings
I Random polynomials
Pemantle Analytic Combinatorics in Several Variables
![Page 21: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/21.jpg)
Running example
Example
Lattice paths to (2n, 2n, 2n) with steps{(2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), (1, 0, 1), (0, 1, 1)}.
F (x , y , z) =1
1− x2 − y 2 − z2 − xy − xz − yz
Pemantle Analytic Combinatorics in Several Variables
![Page 22: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/22.jpg)
Analogy with univariate singularity analysis
1. Find the dominant singularity(ies) and you will know the(limsup) exponential growth rate
2. Behavior of f near the dominant singularity(ies) determinesthe exact asymptotics
Rational multivariate case
F(x) = P(x)/Q(x): singularity is the surface V := {Q = 0}.Carry out same two steps.
Pemantle Analytic Combinatorics in Several Variables
![Page 23: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/23.jpg)
Analogy with univariate singularity analysis
1. Find the dominant singularity(ies) and you will know the(limsup) exponential growth rate
2. Behavior of f near the dominant singularity(ies) determinesthe exact asymptotics
Rational multivariate case
F(x) = P(x)/Q(x): singularity is the surface V := {Q = 0}.Carry out same two steps.
Pemantle Analytic Combinatorics in Several Variables
![Page 24: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/24.jpg)
Analogy with univariate singularity analysis
1. Find the dominant singularity(ies) and you will know the(limsup) exponential growth rate
2. Behavior of f near the dominant singularity(ies) determinesthe exact asymptotics
Rational multivariate case
F(x) = P(x)/Q(x): singularity is the surface V := {Q = 0}.Carry out same two steps.
Pemantle Analytic Combinatorics in Several Variables
![Page 25: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/25.jpg)
Analogy with univariate singularity analysis
1. Find the dominant singularity(ies) and you will know the(limsup) exponential growth rate
2. Behavior of f near the dominant singularity(ies) determinesthe exact asymptotics
Rational multivariate case
F(x) = P(x)/Q(x): singularity is the surface V := {Q = 0}.Carry out same two steps.
Pemantle Analytic Combinatorics in Several Variables
![Page 26: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/26.jpg)
STEP 1: Find dominant singularity
1a. Algebra
1b. Geometry
Pemantle Analytic Combinatorics in Several Variables
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Step 1a: Algebra
I Is the singular surface V smooth?
I If not, what kind of singularities does it have?
Basis
([Q,
∂
∂x1Q, . . . ,
∂
∂xdQ
]);
Answer: [1].
Aha, it’s smooth.
Pemantle Analytic Combinatorics in Several Variables
![Page 28: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/28.jpg)
Step 1a: Algebra
I Is the singular surface V smooth?
I If not, what kind of singularities does it have?
Basis
([Q,
∂
∂x1Q, . . . ,
∂
∂xdQ
]);
Answer: [1].
Aha, it’s smooth.
Pemantle Analytic Combinatorics in Several Variables
![Page 29: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/29.jpg)
Step 1a: Algebra
I Is the singular surface V smooth?
I If not, what kind of singularities does it have?
Basis
([Q,
∂
∂x1Q, . . . ,
∂
∂xdQ
]);
Answer: [1].
Aha, it’s smooth.
Pemantle Analytic Combinatorics in Several Variables
![Page 30: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/30.jpg)
Step 1a: Algebra
I Is the singular surface V smooth?
I If not, what kind of singularities does it have?
Basis
([Q,
∂
∂x1Q, . . . ,
∂
∂xdQ
]);
Answer: [1].
Aha, it’s smooth.
Pemantle Analytic Combinatorics in Several Variables
![Page 31: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/31.jpg)
Step 1a: Algebra
I Is the singular surface V smooth?
I If not, what kind of singularities does it have?
Basis
([Q,
∂
∂x1Q, . . . ,
∂
∂xdQ
]);
Answer: [1].
Aha, it’s smooth.
Pemantle Analytic Combinatorics in Several Variables
![Page 32: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/32.jpg)
Table of contents checklist
1. Introduction
2. Enumeration
3. Univariate asymptotics
4. Complex analysis: univariate saddle integrals
5. Complex analysis: multivariate saddle integrals
6. Symbolic algebra
7. Geometry of minimal points (amoebas)
Pemantle Analytic Combinatorics in Several Variables
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Table of contents checklist
X 1. Introduction
X 2. Enumeration
X 3. Univariate asymptotics
4. Complex analysis: univariate saddle integrals
5. Complex analysis: multivariate saddle integrals
6. Symbolic algebra
7. Geometry of minimal points (amoebas)
Pemantle Analytic Combinatorics in Several Variables
![Page 34: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/34.jpg)
Table of contents checklist
X 1. Introduction
X 2. Enumeration
X 3. Univariate asymptotics
4. Complex analysis: univariate saddle integrals
5. Complex analysis: multivariate saddle integrals
X 6. Symbolic algebra
7. Geometry of minimal points (amoebas)
Pemantle Analytic Combinatorics in Several Variables
![Page 35: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/35.jpg)
Table of contents checklist
X 1. Introduction
X 2. Enumeration
X 3. Univariate asymptotics
X 4. Complex analysis: univariate saddle integrals
5. Complex analysis: multivariate saddle integrals
X 6. Symbolic algebra
7. Geometry of minimal points (amoebas)
Pemantle Analytic Combinatorics in Several Variables
![Page 36: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/36.jpg)
Univariate integrals
∫ ∞−∞
f (t)e−λt2/2 dt =
√2π f (0) λ−1/2
Pemantle Analytic Combinatorics in Several Variables
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Step 1b: Geometry
Next, we use what we know from Step 1a to draw a picture ofthe singularities “nearest to the origin”. In one variable,“nearest” means the least value of |z |. In several variables, wemean those points (x1, . . . , xr ) ∈ V with (|x1|, . . . , |xd |)minimal in the partial order.
Pemantle Analytic Combinatorics in Several Variables
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Step 1b: Geometry
Chapter 7 is the science of determining this set, which is aportion of the boundary of the amoeba of Q. Typically, thisset is a real (d − 1)-dimensional subspace of V . There is ascience to this, which you can read about in Chapter 7.
Often we change to logarith-mic coordinates, in which casethe amoeba looks somethinglike this.
Pemantle Analytic Combinatorics in Several Variables
![Page 39: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/39.jpg)
Step 1b: Geometry
Chapter 7 is the science of determining this set, which is aportion of the boundary of the amoeba of Q. Typically, thisset is a real (d − 1)-dimensional subspace of V . There is ascience to this, which you can read about in Chapter 7.
Often we change to logarith-mic coordinates, in which casethe amoeba looks somethinglike this.
Pemantle Analytic Combinatorics in Several Variables
![Page 40: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/40.jpg)
Step 1b: Geometry
In many natural cases, the coefficients of f are nonnegative. Inthis case there is a Pringsheim theorem telling us that thepostiive real points of V are minimal points.
Example: Q = 1− x2 − y 2 − z2 − xy − xz − yz
Pemantle Analytic Combinatorics in Several Variables
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Step 1b: Geometry
In many natural cases, the coefficients of f are nonnegative. Inthis case there is a Pringsheim theorem telling us that thepostiive real points of V are minimal points.
Example: Q = 1− x2 − y 2 − z2 − xy − xz − yz
Pemantle Analytic Combinatorics in Several Variables
![Page 42: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/42.jpg)
Completing Step 1
Having described the minimal points, we find the dominatingpoint z ∈ V (or x in the amoeba boundary) corresponding tothe asymptotic direction r of interest. It will be the point onthe minimal surface normal to r .
Example: Q = 1− x2− y 2− z2− xy − xz − yz . By symmetry,the point
z∗ :=
(1√6,
1√6,
1√6
)is the dominating point for the diagonal direction. Theexponential growth rate of ar is z−r . Thus,
a2n,2n,2n = (216 + o(1))n .
Pemantle Analytic Combinatorics in Several Variables
![Page 43: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/43.jpg)
Completing Step 1
Having described the minimal points, we find the dominatingpoint z ∈ V (or x in the amoeba boundary) corresponding tothe asymptotic direction r of interest. It will be the point onthe minimal surface normal to r .
Example: Q = 1− x2− y 2− z2− xy − xz − yz . By symmetry,the point
z∗ :=
(1√6,
1√6,
1√6
)is the dominating point for the diagonal direction. Theexponential growth rate of ar is z−r . Thus,
a2n,2n,2n = (216 + o(1))n .
Pemantle Analytic Combinatorics in Several Variables
![Page 44: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/44.jpg)
The fancy stuff: Morse theory
What if the coefficients are not guaranteed to be nonnegativereal numbers?
The Morse theory comes in when the coefficients are of mixedsign or complex and we cannot readily identify the dominatingpoint.
Example (Bi-colored supertrees (DeVries, 2010))
F =P
Q
Q = x5y 2 + 2x2y − 2x3y + 4x + y − 2.
The generating function counts certain combinatorial objectsbut it is only nonnegative in a certain region where theparameters make sense.
Pemantle Analytic Combinatorics in Several Variables
![Page 45: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/45.jpg)
The fancy stuff: Morse theory
What if the coefficients are not guaranteed to be nonnegativereal numbers?
The Morse theory comes in when the coefficients are of mixedsign or complex and we cannot readily identify the dominatingpoint.
Example (Bi-colored supertrees (DeVries, 2010))
F =P
Q
Q = x5y 2 + 2x2y − 2x3y + 4x + y − 2.
The generating function counts certain combinatorial objectsbut it is only nonnegative in a certain region where theparameters make sense.
Pemantle Analytic Combinatorics in Several Variables
![Page 46: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/46.jpg)
The fancy stuff: Morse theory
What if the coefficients are not guaranteed to be nonnegativereal numbers?
The Morse theory comes in when the coefficients are of mixedsign or complex and we cannot readily identify the dominatingpoint.
Example (Bi-colored supertrees (DeVries, 2010))
F =P
Q
Q = x5y 2 + 2x2y − 2x3y + 4x + y − 2.
The generating function counts certain combinatorial objectsbut it is only nonnegative in a certain region where theparameters make sense.
Pemantle Analytic Combinatorics in Several Variables
![Page 47: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/47.jpg)
The fancy stuff: Morse theory
What if the coefficients are not guaranteed to be nonnegativereal numbers?
The Morse theory comes in when the coefficients are of mixedsign or complex and we cannot readily identify the dominatingpoint.
Example (Bi-colored supertrees (DeVries, 2010))
F =P
Q
Q = x5y 2 + 2x2y − 2x3y + 4x + y − 2.
The generating function counts certain combinatorial objectsbut it is only nonnegative in a certain region where theparameters make sense.
Pemantle Analytic Combinatorics in Several Variables
![Page 48: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/48.jpg)
The fancy stuff: Morse theory
What if the coefficients are not guaranteed to be nonnegativereal numbers?
The Morse theory comes in when the coefficients are of mixedsign or complex and we cannot readily identify the dominatingpoint.
Example (Bi-colored supertrees (DeVries, 2010))
F =P
Q
Q = x5y 2 + 2x2y − 2x3y + 4x + y − 2.
The generating function counts certain combinatorial objectsbut it is only nonnegative in a certain region where theparameters make sense.
Pemantle Analytic Combinatorics in Several Variables
![Page 49: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/49.jpg)
The fancy stuff: Morse theory
Finding all possible candidates is an easy algebraiccomputation, producing three possibilities.
The only minimal point isthe rightmost point, butthe dominating point isthe middle point.
This is difficult to deter-mine but you will not usu-ally need to!
Pemantle Analytic Combinatorics in Several Variables
![Page 50: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/50.jpg)
The fancy stuff: Morse theory
Finding all possible candidates is an easy algebraiccomputation, producing three possibilities.
The only minimal point isthe rightmost point, butthe dominating point isthe middle point.
This is difficult to deter-mine but you will not usu-ally need to!
Pemantle Analytic Combinatorics in Several Variables
![Page 51: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/51.jpg)
The fancy stuff: Morse theory
Finding all possible candidates is an easy algebraiccomputation, producing three possibilities.
The only minimal point isthe rightmost point, butthe dominating point isthe middle point.
This is difficult to deter-mine but you will not usu-ally need to!
Pemantle Analytic Combinatorics in Several Variables
![Page 52: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/52.jpg)
Step 1b: Geometry
Summary: computing the minimal points is not trivial, but inmany cases it is not much more than high school geoemtry.
In other words: you don’t need Chapter 7 to get started, andit’s not so bad anyway.
Pemantle Analytic Combinatorics in Several Variables
![Page 53: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/53.jpg)
Table of contents checklist
X 1. Introduction
X 2. Enumeration
X 3. Univariate asymptotics
X 4. Complex analysis: univariate saddle integrals
5. Complex analysis: multivariate saddle integrals
X 6. Symbolic algebra
(X) 7. Geometry of minimal points: amoebas and cones
(X) 9 (parts dealing with finding the dominating point)
Pemantle Analytic Combinatorics in Several Variables
![Page 54: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/54.jpg)
Step 2: classify behavior near singularity and integrate
Use answer from Step 1a (what kind of a point is it?).
Case (i): the dominating point z∗ is a smooth point of V . Wecompute asymptotics in the direction r̂ , resulting in:
ar ∼ C (r̂)n(1−d)/2γn
where
I γ = z−r̂∗I C is computed in an elementary but tedious way from the
partial derivatives of P and Q at z∗ (it is the curvature ofthe minimal surface in logarithmic coordinates).
Pemantle Analytic Combinatorics in Several Variables
![Page 55: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/55.jpg)
Step 2: classify behavior near singularity and integrate
Use answer from Step 1a (what kind of a point is it?).
Case (i): the dominating point z∗ is a smooth point of V . Wecompute asymptotics in the direction r̂ , resulting in:
ar ∼ C (r̂)n(1−d)/2γn
where
I γ = z−r̂∗I C is computed in an elementary but tedious way from the
partial derivatives of P and Q at z∗ (it is the curvature ofthe minimal surface in logarithmic coordinates).
Pemantle Analytic Combinatorics in Several Variables
![Page 56: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/56.jpg)
Step 2: classify behavior near singularity and integrate
Use answer from Step 1a (what kind of a point is it?).
Case (i): the dominating point z∗ is a smooth point of V . Wecompute asymptotics in the direction r̂ , resulting in:
ar ∼ C (r̂)n(1−d)/2γn
where
I γ = z−r̂∗I C is computed in an elementary but tedious way from the
partial derivatives of P and Q at z∗ (it is the curvature ofthe minimal surface in logarithmic coordinates).
Pemantle Analytic Combinatorics in Several Variables
![Page 57: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/57.jpg)
Step 2: classify behavior near singularity and integrate
Use answer from Step 1a (what kind of a point is it?).
Case (i): the dominating point z∗ is a smooth point of V . Wecompute asymptotics in the direction r̂ , resulting in:
ar ∼ C (r̂)n(1−d)/2γn
where
I γ = z−r̂∗
I C is computed in an elementary but tedious way from thepartial derivatives of P and Q at z∗ (it is the curvature ofthe minimal surface in logarithmic coordinates).
Pemantle Analytic Combinatorics in Several Variables
![Page 58: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/58.jpg)
Step 2: classify behavior near singularity and integrate
Use answer from Step 1a (what kind of a point is it?).
Case (i): the dominating point z∗ is a smooth point of V . Wecompute asymptotics in the direction r̂ , resulting in:
ar ∼ C (r̂)n(1−d)/2γn
where
I γ = z−r̂∗I C is computed in an elementary but tedious way from the
partial derivatives of P and Q at z∗ (it is the curvature ofthe minimal surface in logarithmic coordinates).
Pemantle Analytic Combinatorics in Several Variables
![Page 59: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/59.jpg)
Step 2, running example
r̂ = (1, 1, 1)
Q = 1− x2 − y 2 − z2 − xy − xz − yz
z∗ =
(1√6,
1√6,
1√6
)γ = 63/2
C (r̂) =
√3
5π
which leads to
a2n,2n,2n ∼ 216n
[√3
5πn+ O(n−2)
]Pemantle Analytic Combinatorics in Several Variables
![Page 60: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/60.jpg)
Step 2, running example
a2n,2n,2n ∼ 216n√
3/(5πn)
Numerical accuracy: For n = 16, relative error is −0.0068.
Also, we can easily get next term of asymptotics.
The O(n−2) term is169√
3
5625πn2.
Adding this term gives relative error of 0.000023 with n = 16.
“Easily”: SAGE code exists written by A. Raichev + MCW
Pemantle Analytic Combinatorics in Several Variables
![Page 61: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/61.jpg)
Step 2, running example
a2n,2n,2n ∼ 216n√
3/(5πn)
Numerical accuracy: For n = 16, relative error is −0.0068.
Also, we can easily get next term of asymptotics.
The O(n−2) term is169√
3
5625πn2.
Adding this term gives relative error of 0.000023 with n = 16.
“Easily”: SAGE code exists written by A. Raichev + MCW
Pemantle Analytic Combinatorics in Several Variables
![Page 62: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/62.jpg)
Step 2, running example
a2n,2n,2n ∼ 216n√
3/(5πn)
Numerical accuracy: For n = 16, relative error is −0.0068.
Also, we can easily get next term of asymptotics.
The O(n−2) term is169√
3
5625πn2.
Adding this term gives relative error of 0.000023 with n = 16.
“Easily”: SAGE code exists written by A. Raichev + MCW
Pemantle Analytic Combinatorics in Several Variables
![Page 63: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/63.jpg)
Step 2, running example
a2n,2n,2n ∼ 216n√
3/(5πn)
Numerical accuracy: For n = 16, relative error is −0.0068.
Also, we can easily get next term of asymptotics.
The O(n−2) term is169√
3
5625πn2.
Adding this term gives relative error of 0.000023 with n = 16.
“Easily”: SAGE code exists written by A. Raichev + MCW
Pemantle Analytic Combinatorics in Several Variables
![Page 64: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/64.jpg)
Step 2, running example
a2n,2n,2n ∼ 216n√
3/(5πn)
Numerical accuracy: For n = 16, relative error is −0.0068.
Also, we can easily get next term of asymptotics.
The O(n−2) term is169√
3
5625πn2.
Adding this term gives relative error of 0.000023 with n = 16.
“Easily”: SAGE code exists written by A. Raichev + MCW
Pemantle Analytic Combinatorics in Several Variables
![Page 65: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/65.jpg)
Step 2, running example
a2n,2n,2n ∼ 216n√
3/(5πn)
Numerical accuracy: For n = 16, relative error is −0.0068.
Also, we can easily get next term of asymptotics.
The O(n−2) term is169√
3
5625πn2.
Adding this term gives relative error of 0.000023 with n = 16.
“Easily”: SAGE code exists written by A. Raichev + MCW
Pemantle Analytic Combinatorics in Several Variables
![Page 66: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/66.jpg)
Multivariate complex analysis
Incidentally, the math for the multivariate integrals in thesmooth case is simply the multivariate version of saddle pointintegration.
Instead of integrating f (z) exp(−λφ(z)) dz near where φ′
vanishes, we integrate near where ∇φ vanishes:∫f (z) exp(−λφ(z)) ∼
(2π
λ
)d/2
f (0)H−1/2
where H is the determinant of the Hessian matrix of φ.
This is no more difficult, and the statements (which are all youneed) are straightforward.
Time to update the checklist.
Pemantle Analytic Combinatorics in Several Variables
![Page 67: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/67.jpg)
Multivariate complex analysis
Incidentally, the math for the multivariate integrals in thesmooth case is simply the multivariate version of saddle pointintegration.
Instead of integrating f (z) exp(−λφ(z)) dz near where φ′
vanishes, we integrate near where ∇φ vanishes:∫f (z) exp(−λφ(z)) ∼
(2π
λ
)d/2
f (0)H−1/2
where H is the determinant of the Hessian matrix of φ.
This is no more difficult, and the statements (which are all youneed) are straightforward.
Time to update the checklist.
Pemantle Analytic Combinatorics in Several Variables
![Page 68: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/68.jpg)
Multivariate complex analysis
Incidentally, the math for the multivariate integrals in thesmooth case is simply the multivariate version of saddle pointintegration.
Instead of integrating f (z) exp(−λφ(z)) dz near where φ′
vanishes, we integrate near where ∇φ vanishes:∫f (z) exp(−λφ(z)) ∼
(2π
λ
)d/2
f (0)H−1/2
where H is the determinant of the Hessian matrix of φ.
This is no more difficult, and the statements (which are all youneed) are straightforward.
Time to update the checklist.
Pemantle Analytic Combinatorics in Several Variables
![Page 69: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/69.jpg)
Multivariate complex analysis
Incidentally, the math for the multivariate integrals in thesmooth case is simply the multivariate version of saddle pointintegration.
Instead of integrating f (z) exp(−λφ(z)) dz near where φ′
vanishes, we integrate near where ∇φ vanishes:∫f (z) exp(−λφ(z)) ∼
(2π
λ
)d/2
f (0)H−1/2
where H is the determinant of the Hessian matrix of φ.
This is no more difficult, and the statements (which are all youneed) are straightforward.
Time to update the checklist.
Pemantle Analytic Combinatorics in Several Variables
![Page 70: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/70.jpg)
Table of contents checklist
X 1. Introduction
X 2. Enumeration
X 3. Univariate asymptotics
X 4. Complex analysis: univariate saddle integrals
X 5. Complex analysis: multivariate saddle integrals
X 6. Symbolic algebra
(X) 7. Geometry of minimal points: amoebas and cones
(X) 9 (parts dealing with finding the dominating point)
Pemantle Analytic Combinatorics in Several Variables
![Page 71: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/71.jpg)
Step 2: remaining cases go into Part III
Part III addresses:
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I Chapters 12 and 13: wrapping it up (examples and furtherspeculation)
I will not pretened Chapters 10 and 11 are easy, but you willnot need Chapter 11 unless you are lucky enough to run acrossGF’s with an unusual nature.
Pemantle Analytic Combinatorics in Several Variables
![Page 72: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/72.jpg)
Step 2: remaining cases go into Part III
Part III addresses:
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I Chapters 12 and 13: wrapping it up (examples and furtherspeculation)
I will not pretened Chapters 10 and 11 are easy, but you willnot need Chapter 11 unless you are lucky enough to run acrossGF’s with an unusual nature.
Pemantle Analytic Combinatorics in Several Variables
![Page 73: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/73.jpg)
Step 2: remaining cases go into Part III
Part III addresses:
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I Chapters 12 and 13: wrapping it up (examples and furtherspeculation)
I will not pretened Chapters 10 and 11 are easy, but you willnot need Chapter 11 unless you are lucky enough to run acrossGF’s with an unusual nature.
Pemantle Analytic Combinatorics in Several Variables
![Page 74: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/74.jpg)
Step 2: remaining cases go into Part III
Part III addresses:
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I Chapters 12 and 13: wrapping it up (examples and furtherspeculation)
I will not pretened Chapters 10 and 11 are easy, but you willnot need Chapter 11 unless you are lucky enough to run acrossGF’s with an unusual nature.
Pemantle Analytic Combinatorics in Several Variables
![Page 75: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/75.jpg)
Step 2: remaining cases go into Part III
Part III addresses:
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I Chapters 12 and 13: wrapping it up (examples and furtherspeculation)
I will not pretened Chapters 10 and 11 are easy, but you willnot need Chapter 11 unless you are lucky enough to run acrossGF’s with an unusual nature.
Pemantle Analytic Combinatorics in Several Variables
![Page 76: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/76.jpg)
Step 2: remaining cases go into Part III
Part III addresses:
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I Chapters 12 and 13: wrapping it up (examples and furtherspeculation)
I will not pretened Chapters 10 and 11 are easy, but you willnot need Chapter 11 unless you are lucky enough to run acrossGF’s with an unusual nature.
Pemantle Analytic Combinatorics in Several Variables
![Page 77: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/77.jpg)
Interlude: diagonal computation not recommended
Self-intersections such as are in Chapter 10 arise when onecomputes asymtotics of an algebraic d-variate generatingfunction F by embedding it as a diagonal of a rationalfunction.
Unextracting the diagonal, unlike diagonal extraction, is nottoo hard and allows us to extend everything we know aboutrational functions to the algebraic case. Now that you knowabout it, we can check off Chapters 12 and 13.
Pemantle Analytic Combinatorics in Several Variables
![Page 78: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/78.jpg)
Interlude: diagonal computation not recommended
Self-intersections such as are in Chapter 10 arise when onecomputes asymtotics of an algebraic d-variate generatingfunction F by embedding it as a diagonal of a rationalfunction.
Unextracting the diagonal, unlike diagonal extraction, is nottoo hard and allows us to extend everything we know aboutrational functions to the algebraic case. Now that you knowabout it, we can check off Chapters 12 and 13.
Pemantle Analytic Combinatorics in Several Variables
![Page 79: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/79.jpg)
Part III checklist, updated
I X Chapter 9: smooth points
I Chapter 10: interesections of smooth surfaces
I Chapter 11: more complicated local geometry
I XChapters 12 and 13: wrapping it up (examples andfurther speculation)
Pemantle Analytic Combinatorics in Several Variables
![Page 80: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/80.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 81: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/81.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 82: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/82.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 83: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/83.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 84: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/84.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials,
generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 85: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/85.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions,
Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 86: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/86.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and
some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 87: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/87.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 88: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/88.jpg)
What’s left?
Case (ii): Self-intersecting smooth surfaces
Requires theory of iterated residues
Case (iii): More complicated local geometry
Requires theory of hyperbolic polynomials, generalizedfunctions, Leray and Petrovsky cycles and some classicalinverse Fourier transform computations.
These are difficult and we are not going to check them offtoday.
Pemantle Analytic Combinatorics in Several Variables
![Page 89: Analytic Combinatorics in Several Variablespemantle//AofA.pdfThe generating function counts certain combinatorial objects but it is only nonnegative in a certain region where the parameters](https://reader033.fdocuments.in/reader033/viewer/2022061003/60b1eac6bfe22e35d15a11af/html5/thumbnails/89.jpg)
You know what to do!
Pemantle Analytic Combinatorics in Several Variables