ar X iv : m at h / 05 07 23 3 v 1 [ m at h . A G ] 1 2 Ju l 2 00 5 RATIONAL TRANSFORMATIONS OF ALGEBRAIC CURVES AND ELIMINATION THEORY

Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic curve. Introduction The main goal of this research is to describe explicitly an image of an algebraic curve under rational transformation. The simplest and very illustrative case is a rational transformation of a projective line into projective plane. Three homogeneous polynomials in two variables p0, p1 and p2 maps a projective line CP into projective plane CP : (x0, x1) → (p0(x0, x1), p1(x0, x1), p2(x0, x1)) This case was described by N. Kravitsky using the classical elimination theory, see [7]. The image of a projective line is a rational curve. This curve is defined by a polynomial ∆(x0, x1, x2) = det(x0B(p1, p2) + x1B(p2, p0) + x2B(p0, p1)) where B(pi, pj) is the Bezout matrix of polynomials pi and pj. Our original objective was to find an analogue of the constructions of [7] in the general case. This led us to consider elimination theory for pairs of polynomials along an algebraic curve given by a determinantal representation. While our results, as presented in this paper, are for polynomials in two variables, plane algebraic curves, and pairs of operators, the generalization to polynomials in d variables, algebraic curves in the d-dimensional space, and d-tuples of operators should be, for the most part, relatively straightforward. Let us recall the main goal of elimination theory. Given n+1 (nonhomogeneous) polynomials in n variables we want to find necessary and sufficient conditions (in Partially supported by EU-network HPRN-CT-2009-00099(EAGER) , (The Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany), the Israel Science Foundation grant # 8008/02-3 (Excellency Center ”Group Theoretic Methods in the Study of Algebraic Varieties”). 1 2 ALEXANDER SHAPIRO AND VICTOR VINNIKOV terms of the coefficients) for these polynomials to have a common zero (and furthermore to determine the number of common zeroes, counting multiplicities, if they exist), see [9]. In the classical case we consider (nonhomogeneous) polynomials in one variable, p(x) = p0 + p1x+ p2x 2 + · · ·+ pnx n = n