Dividing polynomials when you only know their values

A recent paper by Amiraslani, Corless, Gonzalez-Vega and Shakoori studies polynomial algebra by values, without first converting to another basis such as the monomial basis. In this talk I expand on some details from that paper, namely the method we used to divide (multivariate and univariate) polynomials given only by values. This is a surprisingly valuable operation, and with it one can solve systems of polynomial equations without first constructing a Gröbner basis, or one can compute Gröbner bases if desired. Introduction This paper discusses algorithms for division of polynomials given by values. For the context and philosophy of this approach, see [1]. 1 Univariate division The univariate case is substantially easier than the multivariate case, in that remainders are (however computed) automatically unique, in any basis. We wish to divide the polynomial A(x), given by its values aj , 1 ≤ j ≤ N on the ordered grid xj by the polynomial B(x) given by its values bj on the same grid. We assume deg (A) = n and deg (B) = m are known, and n ≥ m, else the problem is trivial. We seek the values Qj and Rj such that aj = Qjbj + Rj 1 ≤ j ≤ N, (1) where the Qj are the values of a polynomial of degree n−m, whilst the Rj are the values of a polynomial of degree at most m − 1. We have 2N unknowns, and only N equations. We now impose the degree constraints. We could require that derivatives Q(n−m+1)(xj) = 0 and R(xj) = 0 for all xj , which gives us 2N more equations, overconstraining the system. By using the techniques of [1], we may express these derivatives as a linear combination of the values Qj and of the values Rj . However, we may also impose these degree constraints by using differences, which makes the result simpler in the case of equally spaced grids.