Toric Generalized Characteristic Polynomials

We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over an algebraically closed field) of any $n$ by $n$ system of polynomial equations. Since we use the sparse resultant, we thus obtain complexity bounds (for converting any input polynomial system into a multilinear factorization problem) which are close to cubic in the degree of the underlying variety -- significantly better than previous bounds which were pseudo-polynomial in the classical B\'ezout bound. By carefully taking into account the underlying toric geometry, we are also able to improve the reliability of certain sparse resultant based algorithms for polynomial system solving

The computational complexity of the Chow form

We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.Comment: 60 pages, Latex2

Implicitization of rational surfaces using toric varieties

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maximal minor of this matrix. The second method is the toric extension of the method of moving surfaces, and involves finding linear and quadratic relations (syzygies) among the input polynomials. When there are no base points, we show that these can be put together into a square matrix whose determinant is the implicit equation. Its extension to the case where there are base points is also explored.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of moving surfaces. Paper accepted and to appear in Journal of Algebr

The Newton Polytope of the Implicit Equation

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.Comment: 18 pages, 3 figure

Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve