A sparse distributed representation using prime numbers

Abstract

The Fundamental Theorem of Arithmetic (uniqueness of the prime factorisation of positive integers) allows us to represent multivariate polynomials by LISP lists of ordered pairs of numbers. In this representation one can perform all the elementary polynomial arithmetic operations of adding, negating, subtracting and multiplying multivariate polynomials or raising them to non-negative integer powers. The scheme involves the use of an isomorphic image of the ring of polynomials in n variables with rational coefficients. It has the speed and space advantages of Kronecker's trick to transform multivariate polynomials to univariate polynomials. Additional advantages are that the exponents cannot overflow and that the scheme can accommodate terms with negative integer powers.