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