Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012)...
Transcript of Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012)...
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Solving a Real Analogue of Smale’s 17th
Problem
J. Maurice Rojas
Texas A&M University
May 7, 2012
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OutlineMotivation
Sparsity over R
Chamber Cuttings
1 Motivation
2 Sparsity over R
3 Chamber Cuttings
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Applications of Solving Real Polynomial Systems
Maximum Likelihood Estimation in UQ
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Applications of Solving Real Polynomial Systems
Maximum Likelihood Estimation in UQ
Satellite Orbit Design, Geometric Modelling...
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Applications of Solving Real Polynomial Systems
Maximum Likelihood Estimation in UQ
Satellite Orbit Design, Geometric Modelling...
Refined bounds help in complexity theory
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations in n unknowns befound approximately,
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations in n unknowns befound approximately, on the average, in polynomial time with auniform algorithm?
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations in n unknowns befound approximately, on the average, in polynomial time with auniform algorithm?
Major recent progress has come from two near-solutions:[Beltran & Pardo, 2008] and [Burgisser & Cucker, 2010].
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations in n unknowns befound approximately, on the average, in polynomial time with auniform algorithm?
Major recent progress has come from two near-solutions:[Beltran & Pardo, 2008] and [Burgisser & Cucker, 2010].
Resulting complexity bound is
(d+ n
n
)O(1)
,
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations in n unknowns befound approximately, on the average, in polynomial time with auniform algorithm?
Major recent progress has come from two near-solutions:[Beltran & Pardo, 2008] and [Burgisser & Cucker, 2010].
Resulting complexity bound is
(d+ n
n
)O(1)
, but the real case is
of greater importance in certain settings.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Problem/Theorem
Consider f1, . . . , fn∈C[x1, . . . , xn] with maximal degree D.
Smale’s 17th Problem
Can a zero of n complex polynomial equations in n unknowns befound approximately, on the average, in polynomial time with auniform algorithm?
Major recent progress has come from two near-solutions:[Beltran & Pardo, 2008] and [Burgisser & Cucker, 2010].
Resulting complexity bound is
(d+ n
n
)O(1)
, but the real case is
of greater importance in certain settings.Can you go faster over the reals?
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Warm-Up: One Variable, Three Terms
Can you decide whether1 + cxd1 + xD1 (0<d<D)
has 0, 1, or 2 positive roots, using a number of bit operationssub-linear in D + log c?
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Warm-Up: One Variable, Three Terms
You can decide whether1− cx1964181 + x3178111
has 0, 1, or 2 positive roots,
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Warm-Up: One Variable, Three Terms
You can decide whether1− cx1964181 + x3178111
has 0, 1, or 2 positive roots, by checking whether∆0,196418,317811(1,−c, 1) :=196418196418121393121393c317811 − 317811317811
is <0, =0, or >0.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Warm-Up: One Variable, Three Terms
You can decide whether1− cx1964181 + x3178111
has 0, 1, or 2 positive roots, by checking whether∆0,196418,317811(1,−c, 1) :=196418196418121393121393c317811 − 317811317811
is <0, =0, or >0.
...and the preceding condition = checking the sign of196418 log(196418) + 121393 log(121393) + 317811 log(c)− 317811 log(317811),which can be done in polynomial time via Baker’sTheorem on Linear Forms in Logarithms [1967]!
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Warm-Up: One Variable, Three Terms
You can decide whether1− cx1964181 + x3178111
has 0, 1, or 2 positive roots, by checking whether∆0,196418,317811(1,−c, 1) :=196418196418121393121393c317811 − 317811317811
is <0, =0, or >0.
...and the preceding condition = checking the sign of196418 log(196418) + 121393 log(121393) + 317811 log(c)− 317811 log(317811),which can be done in polynomial time via Baker’sTheorem on Linear Forms in Logarithms [1967]!i.e., you can attain complexity logO(1)(Dc)[Bihan, Rojas, Stella, 2009].
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Discriminant Chambers and Liftings
We call any connected component of the complement ofc∈R \ 0 | ∆0,d,D(c)=0
a (reduced) discriminant chamber.
c=0≈1.944526275...
c= 317811196418196418/317811121393121393/317811
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Discriminant Chambers and Liftings
We call any connected component of the complement ofc∈R \ 0 | ∆0,d,D(c)=0
a (reduced) discriminant chamber.
c=0≈1.944526275...
c= 317811196418196418/317811121393121393/317811
1 + xD1 Lower Binomials
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Discriminant Chambers and Liftings
We call any connected component of the complement ofc∈R \ 0 | ∆0,d,D(c)=0
a (reduced) discriminant chamber.
c=0≈1.944526275...
c= 317811196418196418/317811121393121393/317811
1 + xD1 Lower Binomials
./movie1
1− cxd1,−cxd1 + xD1
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835],
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835], but can we attain complexity logO(1)(Dabc)?
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835], but can we attain complexity logO(1)(Dabc)?
Indeed, while f has no more than 8 real roots, computationalalgebra and numerical algebraic geometry do not (yet) give ussuch a speed-up.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835], but can we attain complexity logO(1)(Dabc)?
Indeed, while f has no more than 8 real roots, computationalalgebra and numerical algebraic geometry do not (yet) give ussuch a speed-up. However, you can go faster if you useA-discriminants...
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835], but can we attain complexity logO(1)(Dabc)?
Note: The support of f here is0, 14, 2129, 22013, D
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835], but can we attain complexity logO(1)(Dabc)?
Note: The support of f here is0, 14, 2129, 22013, D
and we
can identify f with a point in R5 (or R3).
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Variable Not So Trivial
Counting the roots of f(x1) :=1 + ax141 + bx2129
1 + cx22013
1 + xD
can be done within (D + log(abc))O(1) bit operations [Sturm,1835], but can we attain complexity logO(1)(Dabc)?
Note: The support of f here is0, 14, 2129, 22013, D
and we
can identify f with a point in R5 (or R3). Also, it makes sense
to say that this f has size log(Dabc)...
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Faster Real Root Counting (One Variable)
Theorem (Ascher, Avendano, Rojas, Rusek, 2012)
For any finite subset A⊂Z of cardinality 1 + k and maximumcoordinate absolute value D, there is a subset SA⊆R
1+k
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Faster Real Root Counting (One Variable)
Theorem (Ascher, Avendano, Rojas, Rusek, 2012)
For any finite subset A⊂Z of cardinality 1 + k and maximumcoordinate absolute value D, there is a subset SA⊆R
1+k withstable log-uniform content 1
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Faster Real Root Counting (One Variable)
Theorem (Ascher, Avendano, Rojas, Rusek, 2012)
For any finite subset A⊂Z of cardinality 1 + k and maximumcoordinate absolute value D, there is a subset SA⊆R
1+k withstable log-uniform content 1 such that, on any input f ∈SA, onecan count exactly the number of positive roots of f
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Faster Real Root Counting (One Variable)
Theorem (Ascher, Avendano, Rojas, Rusek, 2012)
For any finite subset A⊂Z of cardinality 1 + k and maximumcoordinate absolute value D, there is a subset SA⊆R
1+k withstable log-uniform content 1 such that, on any input f ∈SA, onecan count exactly the number of positive roots of f withinO(k18 logD
)time, relative to the BSS model over R.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Faster Real Root Counting (One Variable)
Theorem (Ascher, Avendano, Rojas, Rusek, 2012)
For any finite subset A⊂Z of cardinality 1 + k and maximumcoordinate absolute value D, there is a subset SA⊆R
1+k withstable log-uniform content 1 such that, on any input f ∈SA, onecan count exactly the number of positive roots of f withinO(k18 logD
)time, relative to the BSS model over R.
Furthermore, restricting to inputs in SA with integercoefficients, we can do the same withinO(size(f)1+k + k18 logD
)bit operations.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Faster Real Root Counting (One Variable)
Theorem (Ascher, Avendano, Rojas, Rusek, 2012)
For any finite subset A⊂Z of cardinality 1 + k and maximumcoordinate absolute value D, there is a subset SA⊆R
1+k withstable log-uniform content 1 such that, on any input f ∈SA, onecan count exactly the number of positive roots of f withinO(k18 logD
)time, relative to the BSS model over R.
Furthermore, restricting to inputs in SA with integercoefficients, we can do the same withinO(size(f)1+k + k18 logD
)bit operations. In particular, if
Baker’s refinement of the abc-Conjecture is true, then we canimprove the last bound to size(F )O(1).
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Chambers Can Be Complicated...
./movie2
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Sparsity in General
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Sparsity in General
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. We call F a(real) (n+ k)-nomial n× n system.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Sparsity in General
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. We call F a(real) (n+ k)-nomial n× n system.
For instance, the supports ofx20121 x−1
3 + 4431x
10062 x−1
3 − 1
x20122 x−11 −
√12x10063 x−1
1 − 1x20123 x−1
2 + e46x10061 x−12 − 1
all lie in(0, 0, 0), (2012, 0,−1), (0, 1006,−1), (−1, 2012, 0), (−1, 0, 1006), (0,−1, 2012), (1006,−1, 0)and we thus have a 7-nomial 3× 3 system.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Sparsity in General
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. We call F a(real) (n+ k)-nomial n× n system.
For instance, the supports ofx20121 x−1
3 + 4431x
10062 x−1
3 − 1
x20122 x−11 −
√12x10063 x−1
1 − 1x20123 x−1
2 + e46x10061 x−12 − 1
all lie in(0, 0, 0), (2012, 0,−1), (0, 1006,−1), (−1, 2012, 0), (−1, 0, 1006), (0,−1, 2012), (1006,−1, 0)and we thus have a 7-nomial 3× 3 system.
This system has exactly 8, 144, 865, 727 complex roots but nomore than 124 roots in R
3+.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Sparsity in General
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. We call F a(real) (n+ k)-nomial n× n system.
For instance, the supports ofx20121 x−1
3 + 4431x
10062 x−1
3 − 1
x20122 x−11 −
√12x10063 x−1
1 − 1x20123 x−1
2 + e46x10061 x−12 − 1
all lie in(0, 0, 0), (2012, 0,−1), (0, 1006,−1), (−1, 2012, 0), (−1, 0, 1006), (0,−1, 2012), (1006,−1, 0)and we thus have a 7-nomial 3× 3 system.
This system has exactly 8, 144, 865, 727 complex roots but nomore than 124 roots in R
3+. (See Khovanskii, Sottile, Bates,
Bihan, Rusek...)
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Sparse Size
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. We call F a(real) (n+ k)-nomial n× n system.
For instance, the supports of
F :=
x2D1 x−13 + axD2 x
−13 ± 1
x2D2 x−11 + bxD3 x
−11 ± 1
x2D3 x−12 + cxD1 x
−12 ± 1
all lie in (0, 0, 0), (D,−1, 0), . . . and we thus have a 7-nomial3× 3 system.
This system has exactly 8D3 − 1 complex roots but no morethan 124 roots in R
3+.
size(F ) here is O(log(D) + log(a) + log(b) + log(c)).
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Conjecture 1
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. Let Ω(n, k)denote the maximal number of nondegenerate roots in R
n+ over
all such F .
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Conjecture 1
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. Let Ω(n, k)denote the maximal number of nondegenerate roots in R
n+ over
all such F .
Local Fewnomial Conjecture (real case)
There are absolute constants C1, C2>0 such that for all n, k≥1,we have (n+ k − 1)C1 minn,k−1 ≤ Ω(n, k) ≤ (n+ k − 1)C2 minn,k−1.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Conjecture 1
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. Let Ω(n, k)denote the maximal number of nondegenerate roots in R
n+ over
all such F .
Local Fewnomial Conjecture (real case)
There are absolute constants C1, C2>0 such that for all n, k≥1,we have (n+ k − 1)C1 minn,k−1 ≤ Ω(n, k) ≤ (n+ k − 1)C2 minn,k−1.
True for n=1 [Descartes, 1637] and k=1 [anon].
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Motivating Conjecture 1
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. Let Ω(n, k)denote the maximal number of nondegenerate roots in R
n+ over
all such F .
Local Fewnomial Conjecture (real case)
There are absolute constants C1, C2>0 such that for all n, k≥1,we have (n+ k − 1)C1 minn,k−1 ≤ Ω(n, k) ≤ (n+ k − 1)C2 minn,k−1.
True for n=1 [Descartes, 1637] and k=1 [anon]. Evidence ingeneral comes from [Khovanski, 1980s], [Rojas, 2004], [Bihan &Sottile, 2007], [Bihan, Rojas, Sottile, 2007], and [Avendano,Pebay, Rojas, Rusek, & Thompson, 2012].
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Real Analogue I of Smale’s 17th Problem
Consider f1, . . . , fn∈R[x1, . . . , xn], each having exponentvectors contained in the same set of n+ k points. Let Ω(n, k)denote the maximal number of nondegenerate roots in R
n+
Conjecture (Exact Counting over R)
Suppose we consider random F with maximum exponentcoordinate D. Then there is a uniform algorithm that, in timepolynomial in Ω(n, k) + logD, computes a positive integer that,with high probability, is exactly the number of roots of F with allcoordinates positive.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
2× 2 Trinomial Systems
The discriminant polynomial ∆(a, b) for
F :=
y41 + ax82 − x82y40
x41 + by82 − x40y82
has degree 23206 and coefficients having thousands of digits:hopeless on any computer algebra system...
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
2× 2 Trinomial Systems
The discriminant polynomial ∆(a, b) for
F :=
y41 + ax82 − x82y40
x41 + by82 − x40y82
has degree 23206 and coefficients having thousands of digits:hopeless on any computer algebra system...
...however, the Horn-Kapranov Uniformizationparametrizes the underlying zero set with a one-line formula!
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Horn-Kapranov Uniformization
Succinctly,
F :=
a1y
41 + a2x82 + a3x
82y40
b1x41 + b2y
82 + b3x40y82
,
has discriminant variety ∇A ⊂ P5Cparametrized via
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Horn-Kapranov Uniformization
Succinctly,
F :=
a1y
41 + a2x82 + a3x
82y40
b1x41 + b2y
82 + b3x40y82
,
has discriminant variety ∇A ⊂ P5Cparametrized via
(λBT
)⊙ tA
∣∣ λ∈C2 , t∈(C∗)3
,
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Horn-Kapranov Uniformization
Succinctly,
F :=
a1y
41 + a2x82 + a3x
82y40
b1x41 + b2y
82 + b3x40y82
,
has discriminant variety ∇A ⊂ P5Cparametrized via
(λBT
)⊙ tA
∣∣ λ∈C2 , t∈(C∗)3
,
where A =
0 82 82 41 0 4041 0 40 0 82 820 0 0 1 1 1
and the columns of B are any
basis for the right nullspace of A :=
[1 · · · 1
A
].
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Horn-Kapranov Uniformization
Succinctly,
F :=
a1y
41 + a2x82 + a3x
82y40
b1x41 + b2y
82 + b3x40y82
,
has discriminant variety ∇A ⊂ P5Cparametrized via
(λBT
)⊙ tA
∣∣ λ∈C2 , t∈(C∗)3
,
where A =
0 82 82 41 0 4041 0 40 0 82 820 0 0 1 1 1
and the columns of B are any
basis for the right nullspace of A :=
[1 · · · 1
A
].
So let’s consider the amoeba of ∇A(R), i.e., the image of thereal part under Log| · |...
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Inner/Outer Chambers
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Walls and Chamber Cuttings
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Cutting Complex in Higher Dimensions
./movie3
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Lower Binomial Systems
./movie4
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Sparse Multivariate Polynomial
Theorem (Avendano, Pebay, Rojas, Rusek, Thompson, 2012)
For any finite subset A⊂Zn of cardinality n+ k and maximum
coordinate absolute value D, there is a subset SA⊆FA
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Sparse Multivariate Polynomial
Theorem (Avendano, Pebay, Rojas, Rusek, Thompson, 2012)
For any finite subset A⊂Zn of cardinality n+ k and maximum
coordinate absolute value D, there is a subset SA⊆FA withstable log-uniform content 1
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Sparse Multivariate Polynomial
Theorem (Avendano, Pebay, Rojas, Rusek, Thompson, 2012)
For any finite subset A⊂Zn of cardinality n+ k and maximum
coordinate absolute value D, there is a subset SA⊆FA withstable log-uniform content 1 such that, on any input F ∈SA, onecan count exactly the number of connected components of thepositive zero set of F
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Sparse Multivariate Polynomial
Theorem (Avendano, Pebay, Rojas, Rusek, Thompson, 2012)
For any finite subset A⊂Zn of cardinality n+ k and maximum
coordinate absolute value D, there is a subset SA⊆FA withstable log-uniform content 1 such that, on any input F ∈SA, onecan count exactly the number of connected components of thepositive zero set of F within O
((n+ k)8.5minn+1,k−1+1 logD
)
time, relative to the BSS model over R.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Sparse Multivariate Polynomial
Theorem (Avendano, Pebay, Rojas, Rusek, Thompson, 2012)
For any finite subset A⊂Zn of cardinality n+ k and maximum
coordinate absolute value D, there is a subset SA⊆FA withstable log-uniform content 1 such that, on any input F ∈SA, onecan count exactly the number of connected components of thepositive zero set of F within O
((n+ k)8.5minn+1,k−1+1 logD
)
time, relative to the BSS model over R. Furthermore,restricting to inputs in SA with integer coefficients, we can dothe same within O
(size(F )n+k + (n+ k)8.5minn+1,k−1+1 logD
)
bit operations.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
One Sparse Multivariate Polynomial
Theorem (Avendano, Pebay, Rojas, Rusek, Thompson, 2012)
For any finite subset A⊂Zn of cardinality n+ k and maximum
coordinate absolute value D, there is a subset SA⊆FA withstable log-uniform content 1 such that, on any input F ∈SA, onecan count exactly the number of connected components of thepositive zero set of F within O
((n+ k)8.5minn+1,k−1+1 logD
)
time, relative to the BSS model over R. Furthermore,restricting to inputs in SA with integer coefficients, we can dothe same within O
(size(F )n+k + (n+ k)8.5minn+1,k−1+1 logD
)
bit operations. In particular, if Baker’s refinement of theabc-Conjecture is true, then we can improve the last bound topolynomial in size(F )minn+1,k−1.
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Idea
−2 −1 0
−6
−4
−2
0
2
4
6
x2D + ayD − z
y2D + bzD − x
z2D + cxD − y
PolynomialSystem
−→ Exponent Vectorsyield data structure
−→Coefficients yieldpoint location
problem−→
Optimal StartSystems for
Polyhedral Homotopy
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
Log|∇A| as a surface
...let’s see how the complement of the last line arrangementparametrizes the underlying A-discriminant amoeba...
./movie5
(thanks to Korben Rusek)
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
♥♥♥ Thank you for listening!
...and Happy Birthday Mike!
J. Maurice Rojas A Real Analogue of Smale’s 17th
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OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c.
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 66: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/66.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 67: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/67.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 68: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/68.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.Baker’s Refined abc [1998]: Defineω(n) :=
∑p|n and p prime
1. Then for all such a, b, c,
c=O(logω(abc) Rad(abc)
ω(abc)! Rad(abc)).
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 69: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/69.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.Baker’s Refined abc [1998]: Defineω(n) :=
∑p|n and p prime
1. Then for all such a, b, c,
c=O(logω(abc) Rad(abc)
ω(abc)! Rad(abc)).
abc implies:
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 70: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/70.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.Baker’s Refined abc [1998]: Defineω(n) :=
∑p|n and p prime
1. Then for all such a, b, c,
c=O(logω(abc) Rad(abc)
ω(abc)! Rad(abc)).
abc implies: (1) Effective Falting’s Theorem [Elkies,1991],
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 71: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/71.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.Baker’s Refined abc [1998]: Defineω(n) :=
∑p|n and p prime
1. Then for all such a, b, c,
c=O(logω(abc) Rad(abc)
ω(abc)! Rad(abc)).
abc implies: (1) Effective Falting’s Theorem [Elkies,1991], (2) Effective∗ Roth’s Theorem [Bombieri, 1994;Surroca 2007],
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 72: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/72.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.Baker’s Refined abc [1998]: Defineω(n) :=
∑p|n and p prime
1. Then for all such a, b, c,
c=O(logω(abc) Rad(abc)
ω(abc)! Rad(abc)).
abc implies: (1) Effective Falting’s Theorem [Elkies,1991], (2) Effective∗ Roth’s Theorem [Bombieri, 1994;Surroca 2007], (3)∗ non-existence of Siegel zeroes forcertain L-functions [Granville, 2000].
J. Maurice Rojas A Real Analogue of Smale’s 17th
![Page 73: Solving a Real Analogue of Smale's 17th Problem · Theorem (Ascher, Avendano, Rojas, Rusek, 2012) For any finite subset A⊂Zof cardinality 1+k and maximum coordinate absolute value](https://reader035.fdocuments.in/reader035/viewer/2022070923/5fbb31aca2397f17f0490335/html5/thumbnails/73.jpg)
OutlineMotivation
Sparsity over R
Chamber Cuttings
The abc-Conjecture
Let a, b, c∈N with no common factors and a+ b=c. Also letRad(n) :=
∏p|n and p prime
p.
abc [Masser, Oesterle, 1985]: For any ε>0, and allsuch a, b, c, there is a constant Kε>0 withc<KεRad(abc)
1+ε.Baker’s Refined abc [1998]: Defineω(n) :=
∑p|n and p prime
1. Then for all such a, b, c,
c=O(logω(abc) Rad(abc)
ω(abc)! Rad(abc)).
abc implies: (1) Effective Falting’s Theorem [Elkies,1991], (2) Effective∗ Roth’s Theorem [Bombieri, 1994;Surroca 2007], (3)∗ non-existence of Siegel zeroes forcertain L-functions [Granville, 2000]. Conversely, suitablysharp versions of (1) or (2) imply variants of abc![Surroca 2007, van Frankenhuysen 2007]J. Maurice Rojas A Real Analogue of Smale’s 17th