Using Fermat to solve large polynomial and matrix problems

Abstract

Fermat is an interactive system for mathematical experimentation. It is a super calculator - computer algebra system, in which the basic items being computed can be rational numbers, modular numbers, finite fields, multivariable polynomials, rational functions, or polynomials modulo other polynomials.In Fermat the default "ground ring" F is the field of rational numbers. One may choose to work modulo a specified integer m, thereby changing the ground ring F from Q to Z/m. On top of this may be attached any number of symbolic variables t₁, t₂, . . ., t_n, thereby creating the polynomial ring F[t₁, t₂, . . ., t_n] and its quotient field, the field of rational functions, whose elements are called quolynomials, Further, polynomials p, q, . . . can be chosen to mod out with, creating the quotient ring F(t₁, t₂, . . .) / < p, q, . . .>, whose elements are called polymods. If this is done correctly, finite fields result. Finally, it is possible to allow Laurent polynomials, those with negative as well as positive exponents. Once the computational ring is established in this way, all computations are of elements of this ring.Fermat has extensive built-in primitives for array and matrix manipulations, such as submatrix, sparse matrix, determinant, minors, normalize, column reduce, reduced row echelon, matrix inverse, Smith normal form, and characteristic polynomial. It is consistently faster than some well known computer algebra systems - orders of magnitude faster in some cases.Fermat is a complete programming language. Programs and data can be saved to an ordinary text file that can be read during a later session or read by some other software system.Fermat has solved real problems that other computer algebra systems could not. It is more efficient in both time and space. These problems have come from algebraic topology, group theory, image processing, computational geometry, decision theory, and signal processing.Most recent applications involve solving systems of polynomial equations with the Dixon Resultant technique. I will demonstrate this method at the conference, in particular how I attack the spurious factor problem.Fermat is available for Windows95/98/NT/etc and Mac OS. Fermat for Linux is ready for beta testing.