## arXiv Analytics

### arXiv:1911.02540 [math.PR]AbstractReferencesReviewsResources

#### How many zeros of a random sparse polynomial are real?

Published 2019-11-06Version 1

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 B\"urgisser, Erg\"ur and Tonelli-Cueto showed that the expected number of real zeros of $f$ in such cases is bounded by $O(\sqrt{k} \log k)$. In this work, we improve the bound to $O(\sqrt{k})$ 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 $\Omega(\sqrt{k})$. 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.

Categories: math.PR, cs.CC, math.CA
Related articles: Most relevant | Search more
arXiv:math/0006113 [math.PR] (Published 2000-06-16)
Random polynomials having few or no real zeros
arXiv:0906.1996 [math.PR] (Published 2009-06-10, updated 2010-07-18)
The real zeros of a random algebraic polynomial with dependent coefficients
arXiv:1701.06752 [math.PR] (Published 2017-01-24)
Expected number of critical points of random holomorphic sections over complex projective space