Discriminant and root separation of integral polynomials

2014/07/23 by Götze, Friedrich, Zaporozhets, Dmitry · 1 citation
#11C08 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1407.6388

Abstract

Consider a random polynomial G_Q(x)=ξ_Q,nxⁿ+ξ_Q,n-1x^n-1+...+ξ_Q,0 with independent coefficients uniformly distributed on 2Q+1 integer points \-Q, ..., Q\. Denote by D(G_Q) the discriminant of G_Q. We show that there exists a constant C_n, depending on n only such that for all Q≥ 2 the distribution of D(G_Q) can be approximated as follows sup_{-∞≤ a≤ b≤∞}|ℙ(a≤ \fracD(G_Q)Q^2n-2≤ b)-∫_a^bφ_n(x) dx|≤(C_n)/(log Q), where φ_n denotes the distribution function of the discriminant of a random polynomial of degree n with independent coefficients which are uniformly distributed on [-1,1]. Let Δ(G_Q) denote the minimal distance between the complex roots of G_Q. As an application we show that for any ε>0 there exists a constant δ_n>0 such that Δ(G_Q) is stochastically bounded from below/above for all sufficiently large Q in the following sense ℙ(δ_n1-ε .

Cited by

• Law of large numbers for the discriminant of random polynomials

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