Abstract

Consider a system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ random real polynomial equations in $n$ variables, where each $f_i$ has a prescribed set of exponent vectors described by a set $A\subseteq \mathbb{N}^n$ of cardinality $t$. Assuming that the coefficients of the $f_i$ are independent Gaussians of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by $\frac{1}{2^{n-1}}\binom{t}{n}$.

Keywords

  1. fewnomials
  2. random polynomials
  3. real algebraic geometry
  4. sparsity

MSC codes

  1. Primary
  2. 60D05; Secondary
  3. 14P99

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Information & Authors

Information

Published In

cover image SIAM Journal on Applied Algebra and Geometry
SIAM Journal on Applied Algebra and Geometry
Pages: 721 - 732
ISSN (online): 2470-6566

History

Submitted: 26 November 2018
Accepted: 1 October 2019
Published online: 10 December 2019

Keywords

  1. fewnomials
  2. random polynomials
  3. real algebraic geometry
  4. sparsity

MSC codes

  1. Primary
  2. 60D05; Secondary
  3. 14P99

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : BU 1371 2-2
H2020 European Research Council https://doi.org/10.13039/100010663 : 787840
Einstein Stiftung Berlin https://doi.org/10.13039/501100006188

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