Symbolic Polynomials with Sparse Exponents Stephen M. Watt Ontario Research Centre for Computer Algebra Department of Computer Science, University of Western Ontario London Ontario, CANADA N6A 5B7 watt@uwo.ca Abstract Earlier work has presented algorithms to factor and compute GCDs of symbolic Laurent polyno- mials, that is multivariate polynomials whose exponents are integer-valued polynomials. These earlier algorithms had the problem of high computational complexity in the number of exponent variables and their degree. The present paper solves this problem, presenting a method that preserves the structure of sparse exponent polynomials. 1 Introduction We are interested in the algebra of polynomials whose exponents are not known in advance, but rather are given by integer-valued expressions, for example x 2m 2 +n +3x n y m 3 +1 + 4. In particular, we consider the case where the exponents are integer-valued polynomials with coefficients in Q. One could imagine other models for integer-valued expressions, but this seems sufficiently general for a number of purposes. We call these “symbolic polynomials.” Symbolic polynomials can be related to exponential polynomials [1] and to families of polynomials with parametric exponents [2, 3, 4]. To date, computer algebra systems have only been able to do simple ring operations on symbolic polynomials. They can add and multiply symbolic polynomials, but not much else. In earlier work, we have given a technical definition of symbolic polynomials, have shown that these symbolic polynomials over the integers form a UFD, and have given algorithms to compute GCDs and factor them [5, 6]. These algorithms fall into two families: extension methods, based on the algebraic independence of variables to different monomial powers (e.g. x, x n , x n 2 ,...), and homomorphism methods, based on the evaluation and interpolation of exponent polynomials. There is a problem with these earlier algorithms, however: they become impractical when the exponent polynomials are sparse. Extension methods introduce an exponential number of new variables and ho- momorphism methods require an exponential number of images. We have attempted to address this by performing sparse interpolation of exponents [7, 8], but this leads to impractical factorizations in the image polynomial domain. This paper presents solves these problems. We show a substitution for the extension method that introduces only a linear number of new variables. The resulting polynomials are super-sparse and may be factored by taking images using Fermat’s little theorem, as done by Giesbrecht and Roche [9]. (Indeed, Fermat’s little theorem can be used in a second stage of projection for our homomorphism method, but there combining images is more complicated.) The remainder of the paper is organized as follows: Section 2 recalls a few elementary facts about integer- valued polynomials and fixed divisors. Section 3 summarizes the extension algorithm that we have presented earlier for dense exponents. Section 4 explains why this algorithm is not suitable for the situation when the exponent polynomials are sparse and shows how to deal with this problem. Section 5 presents the extension algorithms adapted to sparse exponents and Section 6 concludes the paper. 1