Phase transition of multivariate polynomial systems

Phase Transition of Multivariate Polynomial Systems

A random multivariate polynomial system with more equations than variables is likely to be unsolvable. On the other hand if there are more variables than equations, the system has at least one solution with high probability. In this paper we study in detail the phase transition between these two regimes, which occurs when the number of equations equals the number of variables. In particular the...

Accelerated Solution of Multivariate Polynomial Systems of Equations

We propose new Las Vegas randomized algorithms for the solution of a square nonde-generate system of equations, with well-separated roots. The algorithms use O(δ 3 n D 2 log(D) log(b)) arithmetic operations (in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal) to approximate all real roots of the system as well as all roots lying in a fix...

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