We show that any matrix-polynomial combination of free noncommutative random variables each having an algebraic law has again an algebraic law. Our result answers a question raised in a recent paper of Shlyakhtenko and Skoufranis.

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN (θ)= ∑N−1 j=0 {αN− j cos( j +1/2)θ + βN− j sin( j +1/2)θ}, where αj and βj , j = 0,1,2, . . . ,N − 1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials w...

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semi-circle law and the Marčenko-Pastur law are special cases. Algebraicity of a ...

We present a randomized polynomial-time algorithm to generate a random integer according to the distribution of norms of ideals at most N in any given number field, along with the factorization of the integer. Using this algorithm, we can produce a random ideal in the ring of algebraic integers uniformly at random among ideals with norm up to N , in polynomial time. We also present a variant of...

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form a0 (n−1 0 )1/2 + a1 (n−1 1 )1/2 x + a2 (n−1 2 )1/2 x2 + · · · + an−1 (n−1 n−1 )1/2 xn−1 where aj , j = 0, 1, 2, . . . , n − 1, are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given...

Algebraic statistics is a recently evolving field, where one would treat statistical models as algebraic objects and thereby use tools from computational commutative algebra and algebraic geometry in the analysis and computation of statistical models. In this approach, calculation of parameters of statistical models amounts to solving set of polynomial equations in several variables, for which ...

For every prime m ≥ 2, we give a family of tautologies that require super-polynomial size constant-depth Frege proofs from Countm axioms, and whose algebraic translations have constant-degree, polynomial size polynomial calculus refutations over Zm. This shows that constant-depth Frege systems with counting axioms modulo m do not polynomially simulate constant-depth Frege systems with counting ...