On the Degree of the GCD of Random Polynomials over a Finite Field

In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, F q is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over F q . Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd , we investigate the probability distribution of the number of rational (i.e., in F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .

Low-Degree Approximation of Random Polynomials

We prove that with “high probability” a random Kostlan polynomial in $$n+1$$ n + 1 many variables and of degree d can be approximated by a polynomial of “low degree” without changing the topology of its zero set on the sphere $$\mathbb {S}^n$$ S n . The dependence between the “low degree” of the approximation and the “high probability” is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree d is isotopic to the zero set of a polynomial of degree $$O(\sqrt{d \log d})$$ O ( d log d ) . The proof is based on a probabilistic study of the size of $$C^1$$ C 1 -stable neighborhoods of Kostlan polynomials. As a corollary, we prove that certain topological types (e.g., curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials.