On the Number of Real Zeros of Random Fewnomials

Bürgisser, Peter; Ergür, Alperen A.; Tonelli-Cueto, Josué

doi:10.1137/18m1228682

Cited by 10 publications

(9 citation statements)

References 23 publications

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“…For example, the bounds were improved for general [16] and particular systems [14]. Moreover, probabilistically, stronger bounds hold [20,72].…”

Section: T 13 (Bbkmentioning

confidence: 99%

Solving sparse polynomial systems using Groebner bases and resultants

Bender

2022

Preprint

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Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gröbner bases in the 60s, there has been a lot of progress in this domain. Moreover, these equations have been employed to model and solve problems from diverse disciplines such as biology, cryptography, and robotics. Currently, we have a good understanding of how to solve generic systems from a theoretical and algorithmic point of view. However, polynomial equations encountered in practice are usually structured, and so many properties and results about generic systems do not apply to them. For this reason, a common trend in the last decades has been to develop mathematical and algorithmic frameworks to exploit specific structures of systems of polynomials.Arguably, the most common structure is sparsity; that is, the polynomials of the systems only involve a few monomials. Since Bernstein, Khovanskii, and Kushnirenko's work on the expected number of solutions of sparse systems, toric geometry has been the default mathematical framework to employ sparsity. In particular, it is the crux of the matter behind the extension of classical tools to systems, such as resultant computations, homotopy continuation methods, and most recently, Gröbner bases. In this work, we will review these classical tools, their extensions, and recent progress in exploiting sparsity for solving polynomial systems.

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“…For example, the bounds were improved for general [16] and particular systems [14]. Moreover, probabilistically, stronger bounds hold [20,72].…”

Section: T 13 (Bbkmentioning

confidence: 99%

Solving sparse polynomial systems using Groebner bases and resultants

Bender

2022

Preprint

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“…Further, the proof that all the roots are concentrated around 1 follows from the analysis of an approximation of the Edelman-Kostlan integral. This approximation which is inspired by the one used in [2] makes the analysis of the integral simpler.…”

Section: Proof Ideasmentioning

confidence: 99%

“…In [2], they get around this difficulty by upper bounding the integral. This is achieved by ignoring the negative term of the numerator and through some elementary norm inequalities leads to the O( √ k log k) bound.…”

Section: Preliminariesmentioning

confidence: 99%

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How many zeros of a random sparse polynomial are real?

Jindal¹,

Pandey²,

Shukla³

et al. 2019

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We investigate the number of real zeros of a univariate k-sparse polynomial f over the reals, when the coefficients of f come from independent standard normal distributions. Recently Bürgisser, Ergür and Tonelli-Cueto showed that the expected number of real zeros of f in such cases is bounded by O( √ k log k). In this work, we improve the bound to O( √ k) and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by Ω( √ k). Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of f in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the O(log n) bound on the expected number of real zeros of a dense polynomial of degree n with coefficients coming from independent standard normal distributions.

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“…(t − n)! for the number of positive roots was proved in [BET19], using independent real Gaussians for the coefficients. It should also be noted that counting roots on coordinate subspaces quickly abuts #P-hardness, already for n × n binomial systems: See [CD07,Mon20] and Remark 2.9 in Section 2.2 below.…”

Section: Introductionmentioning

confidence: 99%

Counting Real Roots in Polynomial-Time for Systems Supported on Circuits

Rojas

2020

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Suppose A = {a 1 , . . . , a n+2 } ⊂ Z n has cardinality n + 2, with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = (f 1 , . . . , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log(dH).We prove Theorem 1.1 in Section 5.1, based mainly on Algorithms 4.1 and 4.3 from Section 4. A central ingredient is diophantine approximation.The genericity condition for counting roots in (R * ) n and R n + , when t = n + 2, is the nonsingularity of a collection of sub-matrices of [c i,j ], depending solely on the support of F , and

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On the Number of Real Zeros of Random Fewnomials

Cited by 10 publications

References 23 publications

Solving sparse polynomial systems using Groebner bases and resultants

Solving sparse polynomial systems using Groebner bases and resultants

How many zeros of a random sparse polynomial are real?

Counting Real Roots in Polynomial-Time for Systems Supported on Circuits

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