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Let  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 1 ≤  ≤  the polynomial   can be evaluated using  arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family  is in a suitable sense . We construct a database , supported by an algebraic computation tree, such that for each  the query for the signs of  (), . . . ,  () can be answered using  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.

Evaluating geometric queries using few arithmetic operations