A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.

One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment.

Abstract. When factoring linear partial differential systems with a finite-dimensional solution space or analyzing symmetries of nonlinear ode’s, we need to look for rational solutions of certain nonlinear pde’s. The nonlinear pde’s are called Riccati-like because they arise in a similar way as Riccati ode’s. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system {∂xu + u 2 + a1u+a2v+a3, ∂yu+uv+b1u+b2v+b3, ∂xv+uv+c1u+c2v+c3, ∂yv+v 2 +d1u+d2v+d3}, where a1... d3 are rational functions of x and y.

Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.

Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate information concerning the Galois group of P over k 0 (x) as well as over k 0 (x).

We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in sub-quadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem. Key words: Polynomial factorization, Hensel lifting, Bertini’s irreducibility theorem.

We present a comprehensive algorithm to construct a topologically correct triangulation of the real affine part of a rational parametric surface with few restrictions on the defining rational functions. The rational functions are allowed to be undefined on domain curves ( pole curves) and at certain special points (base points), and the surface is allowed to have nodal or cuspidal self-intersections. We also recognize that for a complete display some real points on the parametric surface may be generated only by complex parameter values, and that some finite points on the surface may be generated only by infinite parameter values; we show how to compensate for these conditions. Our techniques for handling these problems have applications in scientific visualization, rendering non-standard NURBS, and in finite-element mesh generation. 1 Introduction Points on a parametric surface patch can be generated by sampling the parametric functions over some region of the parameter domain. Bec...

The set of solutions to a collection of polynomial equations is referred to as an algebraic set. An algebraic set that cannot be represented as the union of two other distinct algebraic sets, neither containing the other, is said to be irreducible. An irreducible algebraic set is also known as an algebraic variety. This paper deals with geometric computations with algebraic varieties. The main results are algorithms to (1) compute the degree of an algebraic variety, (2) compute the rational parametric equations (a rational map from points on a hyperplane) for implicitly de ned algebraic varieties of degrees two and three. These results are based on sub-algorithms using multi-polynomial resultants and multi-polynomial remainder sequences for constructing a one-to-one projection map of an algebraic variety to a hypersurface of equal dimension, as well as, an inverse rational map from the hypersurface to the algebraic variety. These geometric computations arise naturally in geometric modeling, computer aided design, computer graphics, and motion planning, and have been used in the past for special cases of algebraic varieties, i.e. algebraic curves and surfaces.

We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably square-free, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.

Abstract: Our problem is to decompose a positive dimensional solution set of a polynomial system into irreducible components. This solution set is represented by a witness set, obtained by intersecting the set with random linear slices of complementary dimension. Points on the same irreducible components are connected by path tracking techniques applying the idea of monodromy. The computation of a linear trace for each component certifies the decomposition. This decomposition method exhibits a good practical performance on solution sets of relatively high degrees defined by systems of low degree polynomials.