Newton Polytopes and Witness Sets

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface H in Cn using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straightline program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes the Newton polytope of H using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when H is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face. Introduction While a hypersurfaceH in C is always defined by the vanishing of a single polynomial f , we may not always have access to the monomial representation of f . This occurs, for example, when H is the image of a map or if f is represented as a straight-line program, and it is a well-understood and challenging problem to determine the polynomial f whenH is represented in this way. Elimination theory gives a symbolic method based on Gröbner bases that can determine f from a representation of H as the image of a map or as a discriminant [8]. Such computations require that the map be represented symbolically, and they may be infeasible for moderately-sized input. The set of monomials in f , or more simply the convex hull of their exponent vectors (the Newton polytope of f), is an important combinatorial invariant of the hypersurface. The Newton polytope encodes asymptotic information about H and determining it from H is a step towards determining the polynomial f . For example, numerical linear algebra [7, 13] may be used to find f given its Newton polytope. Similarly, the Newton polytope of an image of a map may be computed from Newton polytopes of the polynomials defining the map [12, 14, 15, 31, 32], and computed using algorithms from tropical geometry [33]. We propose numerical methods to compute the Newton polytope of f in two cases when f is not known explicitly. We first show how to compute the Newton polytope when we are able to evaluate f . This occurs, for example, if f is represented as a straight-line program or as a determinant (neither of which we want to expand as a sum of monomials), or perhaps as a compiled program. For the other case, we suppose that f defines a 1991 Mathematics Subject Classification. 14Q15, 65H10.