ar X iv : a lg - g eo m / 9 60 90 03 v 1 6 S ep 1 99 6 Polar Varieties and Efficient Real Equation Solving : The Hypersurface Case

ar X iv : a lg - g eo m / 9 60 80 25 v 1 2 2 A ug 1 99 6 COUNTING PLANE CURVES OF ANY GENUS

v 1 6 S ep 1 99 6 Polar Varieties , Real Equation Solving and Data - Structures : The hypersurface case ∗

In this paper we apply for the first time a new method for multivariate equation solving which was developed in [18], [19], [20] for complex root determination to the real case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of [18], [19], [20] yields a new method for s...

ar X iv : a lg - g eo m / 9 50 90 03 v 1 1 S ep 1 99 5 Generators for Symbolic Powers of Ideals

L et I be an ideal, homogeneous with respect to the usual grading, in a polynomial ring R = k[x0, . . . , xn] in n+ 1 variables (over an algebraically closed field k). Denote the graded component of I of degree d by Id, and likewise the k-vector space of homogeneous forms of R of degree d by Rd. Since I is a graded R-module, we have k-linear maps μd,i : Id ⊗ Ri → Id+i given for each i and d by ...

ar X iv : a lg - g eo m / 9 60 30 11 v 1 1 4 M ar 1 99 6 The Hilbert Schemes of Degree Three Curves are Connected

In this paper we show that the Hilbert scheme H(3, g) of locally Cohen-Macaulay curves in P of degree three and genus g is connected. This is achieved by giving a classification of these curves, determining the irreducible components of H(3, g), and giving certain specializations to show connectedness. As a byproduct, we find that there are curves which lie in the closure of each irreducible co...

ar X iv : a lg - g eo m / 9 70 60 04 v 1 1 0 Ju n 19 97 NUMERICAL SCHUBERT CALCULUS

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gröbner basis for the Plücker ideal of the G...