Roots of random polynomials over a finite field
This is an interesting work on the number of roots of polynomials with random coefficients. The paper generalizes the classical and well known results of \textit{M. Kac} [Bull. Am. Math. Soc. 49, 314--320 (1943; Zbl 0060.28602)] in which he obtained the first asymptotic value for the expected number of real zeros of an algebraic polynomial \(\sum_{j=0}^n a_j x^j\) with random coefficient \(a_j\). This paper instead considers the ring of polynomials over a finite field and studies the number of its roots lying in the finite field. It is shown that this number asymptotically tends to a Poisson distribution with parameter one. Although results are mainly for the normalized polynomials, more general cases are also discussed.
scientific article; zbMATH DE number 5155527