Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Diagonal asymptotics for symmetric rational functions via ACSV

We consider asymptotics of power series coefficients of rational functions of the form 1/Q where Q is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV someti...

Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorit...

Numerical solution of multivariate polynomial systems by homotopy continuation methods

pn(xi,...,xn) = 0 for x = (x\,... ,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Grobner bas...

An introduction to the numerical analysis of multivariate polynomial systems

Comparison of Algorithms for Multivariate Rational Approximation

Let F be a continuous real-valued function defined on the unit square [—1,11 x [-1, 1]. When developing the rational product approximation to F, a certain type of discontinuity may arise. We develop a variation of a known technique to overcome this discontinuity so that the approximation can be programmed. Rational product approximations to F have been computed using both the second algorithm o...