Structure and Eecient Computation of Multiplication Tables and Associated Quadratic Forms 1 Domain and Motivation 2 Multiplication Table 2.1 Characterization

We characterize the multiplication table of an algebra with a multiplicative unity and derive properties of multiplication tables and associated quadratic forms, which allow eecient computation. We compare several ways of computing a multiplication table and associated quadratic forms. By index permutation the complexity of the computation time of associated quadratic forms can be reduced. Exploiting the structure concerning equal entries in the multiplication table gives a further speedup. We apply our results to the case of the factor ring K x]=I and give speedups in the appendix. Let A be a nite dimensional vector space over a eld K and let be a commutative and associative operation ("multiplication") such that (A;) becomes a commutative algebra (associativity is silently assumed here). Furthermore let have a unity ("1") throughout. Instead of a b, for a; b 2 A, we shall simply write ab. Given a basis V = fv 1 ; : : :v d g of A the operation induces a multiplication table w k ij ; 1 i; j; k d; given by v i v j = d X k=1 w k ij v k : (1) In the following we will omit the symbol P for summation. Instead an index appearing both as super-index and subindex will indicate summation over this index (Einstein sum convention, tensor notation). The range of the index will here always be d, the dimension of A. Hence (1) can be written simply as v i v j = w k ij v k. Any such algebra is isomorphic to a factor ring A := K x]=I, where x is an abbreviation for x 1 ; : : :x r and I a zero-dimensional ideal (see the appendix for a proof). The need to compute the multiplication table and associated quadratic forms for a factor ring comes from an algorithm counting common real zeros of a polynomial system ((Pederson et al. 94]). Since is commutative and associative and has a multiplicative unity, for the multiplication table several properties hold. Actually a multiplication table can be characterized by the following lemma. Lemma 2.1 A three dimensional matrix w k ij ; i; j; k = 1 : : :d; is a multiplication table of an algebra (A;) with a multiplicate unity, if and only if the following properties hold: 1. w k ij = w k ji ,