##### Sesión Álgebra Computacional y Conmutativa

Diciembre 14, 12:00 ~ 12:20

## ON BEZOUT INEQUALITIES FOR NON-HOMOGENEOUS POLYNOMIAL IDEALS

### Heintz, Joos

Motivated by the aim to formulate and prove an idealistic version of Bézout's Theorem (see \cite{vogel}) and by applications to transcendence theory (see \cite{MW,Bro}), the notion of degree of \emph{homogeneous} polynomial ideals became intensively studied. In general this work relied on the notion of degree of homogeneous polynomial ideals based on the Hilbert polynomial. In this paper we propose an alternative and self-contained approach for non-homogeneous polynomial ideals.\\ We introduce a ``workable'' notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic Bézout Inequalities for the sum of two ideals in terms of this notion of degree. However it turns out, that, due to the presence of embedded primes, a Bézout Inequality in completely intrinsic terms (depending only on the degrees of the two given ideals) is unfeasable. This differs from the purely geometric point of view, where such a Bézout inequality is known from a long time (see \cite{Heintz83,fulton,vogel}). Hence in some place the degrees of generators of at least one of the ideals comes into play and our main Bézout Inequality for non-homogeneous ideals will be of this mixed type. The result itself is new although it can also be derived from its homogeneous counterpart. We then apply this inequality in order to prove an affine version (with refined bounds) of a celebrated result by D.W. Masser and G. W\"ustholz in \cite{MW}. Finally, we arrive to an important algorithmic application of our theoretical results:\\ We design a nearly optimal probabilistic algorithm which computes the degree of a non-homogeneous equidimensional ideal given by generators. In particular, there is no reference to Hilbert polynomials. In the case of a ground field with efficient factorisation the complexity of this algorithm is polynomial in the degree of the ideal, the syntactical length of the description of the generators and the complexity of performing a Hironaka division with prescribed bounds on the polynomials involved. This is the main algorithm outcome of this manuscript). \begin{footnote} {Research was partially supported by the following Argentinean, Iranian and Spanish Grants: UBACyT 20020130100433BA, PICT-2014-3260, IPM No.95550420, MTM2014-55262-P.}\end{footnote} \begin{thebibliography}{00} {\small \bibitem{Bro} D. Brownawell, W. Masser, \emph{Multiplicity estimates for analytic functions II}. Duke Math. J. \textbf{47} (1980), 273-295. \bibitem{fulton} W. Fulton, \emph{``Intersection Theory"}. Springer-Verlag, 1984. \bibitem{Heintz83} J. Heintz, \emph{Definability and Fast Quantifier Elimination in Algebraic Closed Fields}. \bibitem{MW} D.W. Masser, G. W\"ustholz, \emph{Fields of Large Transcendence Degree Generated by Values of Elliptic Functions.} Inventiones Mathematicae \textbf{72} (1983), 407-464. \bibitem{vogel} W. Vogel, \emph{Lectures on results on Bezout's Theorem}. Tata Lect. Notes, Bombay, vol.\textbf{74}. Springer-Verlag, 1984.} \end{thebibliography}

Autores: Heintz, Joos / Amir, Hashemi / Pardo, Luis Miguel / Solernó, Pablo.