On the Sum of Square Roots of Polynomials and Related Problems

Abstract

The sum of square roots over integers problem is the task of deciding the sign of a nonzero sum, S = ∑ i=1 n δ i · √ a i , where δ i ∈ {+1, −1} and a i ’s are positive integers that are upper bounded by N (say). A fundamental open question in numerical analysis and computational geometry is whether ∣ S ∣ ≥ 1/2 (n ·log N)O(1) when S ≠ 0. We study a formulation of this problem over polynomials. Given an expression S = ∑ i=1 n c i · √ f i ( x ), where c i ’s belong to a field of characteristic 0 and f i ’s are univariate polynomials with degree bounded by d and f i (0)≠0 for all i , is it true that the minimum exponent of x that has a nonzero coefficient in the power series S is upper bounded by ( n · d ) O(1) , unless S = 0? We answer this question affirmatively. Further, we show that this result over polynomials can be used to settle (positively) the sum of square roots problem for a special class of integers: Suppose each integer a i is of the form, a i = X d i + b i1 X di−1 +...+ b idi , d i > 0, where X is a positive real number and b ij ’s are integers. Let B = max ({∣ b ij ∣} i, j , 1) and d = max i { d i }. If X > ( B + 1) (n·d)O(1) then a nonzero S = ∑ i=1 n δ i · √ a i is lower bounded as ∣ S ∣ ≥ 1/ X (n·d)O(1) . The constant in O (1), as fixed by our analysis, is roughly 2. We then consider the following more general problem. Given an arithmetic circuit computing a multivariate polynomial f ( X ) and integer d , is the degree of f ( X ) less than or equal to d ? We give a coRP PP -algorithm for this problem, improving previous results of Allender et al. [2009] and Koiran and Perifel [2007].