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Solving Systems of Equations and Inequalities over Specified Domains

Adam Strzebonski
Organization: Wolfram Research, Inc.
Department: Kernel Technology

2004 Wolfram Technology Conference
Conference location

Champaign IL

Mathematica 5 contains new functionality that allows users to solve, possibly quantified, systems of equations and inequalities with domain restrictions on the variables. The function Reduce eliminates quantifiers and provides an explicit description of the entire solution set. The function FindInstance allows users to find a specified finite number of solutions (if they exist). The function Resolve eliminates quantifiers producing an equivalent quantifier-free system. In the upcoming version of Mathematica the functionality has been extended to handle several new classes of systems. The classes of systems that we can solve in full generality include quantified systems of polynomial equations and inequations in complex variables or in integers modulo n, quantified systems of polynomial equations and inequalities in real or complex variables, systems of linear equations and inequalities in integer, real, or complex variables, systems of polynomial congruences, and quantified systems of equations and inequalities in real variables containing real algebraic functions. Complex systems containing algebraic functions can be solved up to branch cut dependent equalities. We can also solve several classes of transcendental systems that can be decomposed into algebraic and simple transcendental systems. Polynomial systems in integer variables are not solvable in general (by Matiyasevich’s solution of Hilbert’s 10th problem), however, we have implemented algorithms for solving several classes of such systems. Systems involving piecewise functions can be solved as long as the piecewise functions are constructed with conditions and functions of solvable types. In this talk I will show some examples, discuss what we can and what we can not solve, and what algorithms we use.

*Applied Mathematics > Optimization
*Mathematics > Algebra > Polynomials
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ReduceTalk.nb (2 MB) - Presentation notebook [for Mathematica 5.1]