On the Sum of Square Roots of Polynomials and Related Problems

• Authors: Neeraj Kayal;Chandan Saha
• Venue: ACM Transactions on Computation Theory (TOCT)
• Year: 2012

The sum of square roots over integers problem is the task of deciding the sign of a nonzero sum, S = ∑i=1n δi · √ai, where δi ∈ {+1, −1} and ai’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=1n ci · √fi(x), where ci’s belong to a field of characteristic 0 and fi’s are univariate polynomials with degree bounded by d and fi(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 ai is of the form, ai = Xdi + bi1Xdi−1 +...+ bidi, di 0, where X is a positive real number and bij’s are integers. Let B = max ({∣bij∣}i, j, 1) and d = maxi{di}. If X (B + 1)(n·d)O(1) then a nonzero S = ∑i=1n δi · √ai 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 coRPPP-algorithm for this problem, improving previous results of Allender et al. [2009] and Koiran and Perifel [2007].