Black-Box Polynomial Resultants

A black-box polynomial is a multivariate polynomial that is represented by a program that evaluates the polynomial at an arbitrary point supplied as input. The paper describes an algorithm for constructing a black box for the resultant of two black-box polynomials. The only computationally nontrivial step in the construction is that which determines the degrees of the input black boxes in the variable being eliminated; if those degrees are known, then the black-box resultant can be constructed in a bounded amount of time. Let N be an upper bound for the degrees of the input polynomials. The black-box resultant can be evaluated at an arbitrary point with O(N) calls to the input black boxes and O(N²) arithmetic operations.