Exact Computat ion Using Approximate Gröbner Bases

Momentum Distribution and Charge Density in Solid-State Theory *

We present a discussion of a number of conceptual and methodological aspects associated with the theoretical characterization and computat ion of charge densities and momentum distributions in solids. The main ambition has been to stress properties that both exact and approximate quantities must possess. We have also attempted to point out conceptual and computat ional trends which would seem t...

Approximate Gröbner Bases and Overdetermined Algebraic Systems

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to accuracy and precision from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we...

Applying Gröbner Bases To“solve” Solvable Polynomials

This paper is designed to introduce the reader to Gröbner bases as well as demonstrate how they may applied in developing algorithms that“solve” solvable polynomials. Specifically, I will use these bases along with some basic invariant theory in order to derive a general formula for the roots of a cubic polynomial with rational coefficients and describe how to compute the exact roots of a quint...

Automatic Geometric Theorem Proving using Gröbner Bases

Counting Zeros over Finite Fields Using Gröbner Bases