Polynomial maps with maximal multiplicity and the special closure
For a polynomial \(F:\mathbb C^n\to\mathbb C^n\) one defines \(\mu(F)\) to be the dimension of the algebra obtained by quotienting out polynomials on \(\mathbb C^n\) by the components of \(F\). This number can also be interpreted as the expected number of solutions of the system \(F=0\). It is known that \(\mu(F)\) is bounded above by \(n! \mathrm{Vol}(\widetilde{\Gamma}_+(F))\), where \(\widetilde{\Gamma}_+(F)\) is a convex polyhedron obtained when studying which monomials appear with non-0 coefficients in \(F\). The main result of this paper is that in this estimate we have equality if an only if \(F\) is Newton non-degenerate at infinity -- an important condition on polynomials, involving the convex geometry of the faces of \(\widetilde{\Gamma}_+(F)\). In the last section the authors introduce and study the notion of \(F\) being non-degenerate with respect to a ``global Newton polyhedron''. In Theorem 4.9 these maps are proved to be the ones for which a certain \(\mu\)-vs-Volume estimate is sharp.
scientific article; zbMATH DE number 7031251
scientific article