Computation of traces by means of resultants

GABRIELA JERONIMO

Colonia, Uruguay

Congreso; XVI Coloquio Latinoamericano de Álgebra; 2005

Traces in finite dimensional algebras play a fundamental role in Commutative Algebra and Algebraic Geometry. Recent applications of traces include the evaluation of symmetric functions, the effective Nullstellensatz and algorithms for solving polynomial systems. Let $mathbb{K}$ be a field of characteristic zero. If $mathcal{I}subset mathbb{K}[x_1,dots, x_n]$ is a zero-dimensional ideal, a rational function $rin mathbb{K}(x_1,dots, x_n)$  whose denominator is not a zero divisor in $mathbb{K}[x_1,dots, x_n] /mathcal{I}$ induces a $mathbb{K}$-linear map  $amapsto rcdot a$ in this finite dimensional $mathbb{K}$-algebra. We will focus on the computation of the traces of these linear maps. We will discuss the case where $mathcal{I}$ is given by a generic complete intersection in the torus. We will exhibit a formula for the trace  as a rational function in the coefficients of $r$ and the coefficients of a given generator set of $mathcal{I}$, in which both the numerator and the denominator are described in terms of sparse resultants. Finally, for complete intersections in the affine space without zeroes at the infinity, we will show how to recover some classical results about multidimensional residues by means of our resultant-based formulas for traces. (Joint work with Carlos D´Andrea.)