How many zeros of a random sparse polynomial are real?

Jindal, Gorav; Pandey, Anurag; Shukla, Himanshu; Zisopoulos, Charilaos

doi:10.1145/3373207.3404031

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…The bounds for the number of real roots of random but sparse polynomials were provided in [37]. A numerical method for efficiently finding the zeros of complex-valued polynomials of very large orders was developed in [38].…”

Section: Probability Density Transformationsmentioning

confidence: 99%

Polynomial Distributions and Transformations

Loskot

2023

Mathematics

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Polynomials are common algebraic structures, which are often used to approximate functions, such as probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer great modeling flexibility and mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the intervals of support for some parameter values; their parameter numbers are usually much larger than for canonical distributions, and the interval of support must be finite. Hence, polynomial distributions are defined here assuming three forms of a polynomial function. Transformations and approximations of distributions and histograms by polynomial distributions are also considered. The key properties of the polynomial distributions are derived in closed form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. A goodness-of-fit measure is proposed to determine the best order of the approximating polynomial. Numerical examples include the estimation of parameters of the polynomial distributions and generating polynomially distributed samples.

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Section: Probability Density Transformationsmentioning

confidence: 99%

Polynomial Distributions and Transformations

Loskot

2023

Mathematics

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“…We finish with a result on the typical location of the zeros. It is well known that for random real polynomials, the positive reals zeros x tend to accumulate around 1: see [12] for the dense and [19] for the sparse case. This means that w = log x accumulates around 0.…”

Section: Conjecturementioning

confidence: 99%

“…The univariate case and a conjecture. The univariate case (n = 1) was settled by Jindal et al [19]. They showed that for any subset S ⊆ R of cardinality t, we have…”

Section: Introductionmentioning

confidence: 99%

Real zeros of mixed random fewnomial systems

Bürgisser¹

2023

Preprint

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Consider a system f 1 (x) = 0, . . . , fn(x) = 0 of n random real polynomials in n variables, where each f i has a prescribed set of exponent vectors in a set A i ⊆ Z n of cardinality t i , whose convex hull is denoted P i . Assuming that the coefficients of the f i are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most (2π) − n 2 V 0 (t 1 − 1) . . . (tn − 1). Here V 0 denotes the number of vertices of the Minkowski sum P 1 + . . . + Pn. We also derive a better bound in the unmixed case where all supports A i are equal, improving upon Bürgisser et al. (SIAM J. Appl. Algebra Geom. 3(4), 2019). All arguments equally work for real exponent vectors.

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Solving Sparse Polynomial Systems using Gröbner Bases and Resultants

Bender

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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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. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; Hybrid symbolic-numeric methods; Symbolic calculus algorithms; Equation and inequality solving algorithms.

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How many zeros of a random sparse polynomial are real?

Cited by 3 publications

References 16 publications

Polynomial Distributions and Transformations

Polynomial Distributions and Transformations

Real zeros of mixed random fewnomial systems

Solving Sparse Polynomial Systems using Gröbner Bases and Resultants

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