Evaluating geometric queries using few arithmetic operations

Let \({\mathcal{P}:=(P_1,\ldots,P_s)}\) be a given family of n-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by d and h, respectively. Suppose furthermore that for each 1 ≤ i ≤ s the polynomial Pi can be evaluated using L arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family \({\mathcal{P}}\) is in a suitable sense generic. We construct a database \({\mathcal{D}}\), supported by an algebraic computation tree, such that for each \({x\in [0,1]^n}\) the query for the signs of P1(x), . . . , Ps(x) can be answered using \({h d^{\mathcal{O}(n^2)}}\) comparisons and nL arithmetic operations between real numbers. The arithmetic-geometric tools developed for the construction of \({\mathcal{D}}\) are then employed to exhibit example classes of systems of n polynomial equations in n unknowns whose consistency may be checked using only few arithmetic operations, admitting, however, an exponential number of comparisons.