Speaker
Description
In the talk, I will go through our recent work regarding the Smith form of the Sylvester and Bézout resultant matrices. The partial multiplicities associated to the eigenvalue of a polynomial matrix tell us about the conditioning of computing the eigenvalue. Since the eigenvalues of the resultant matrices are the roots of the system, the partial multiplicities are connected to the stability of solving the polynomial system. We have also attempted to keep the assumptions as general as possible, and as a result, our coefficient field can be any field.
For bivariate polynomials $f,g \in \mathbb{K}[x,y]$, we make a connection between the partial multiplicities of the eigenvalues of the resultant matrices and the structure of the dual space of the zero-dimensional ideal $\langle f,g \rangle$. We show this by explicitly constructing the root polynomials for the matrix $S(y)$ using a Gauss basis of the dual space, introduced in [3M96].
In the talk, I aim to go through the general idea of the proof, as well as intuitively introduce the results through various examples. We first consider the case where $f$ and $g$ have coprime leading coefficients, as then $\det S(y) = \mathrm{Res}(f,g)$. Then, via means of a Möbius transformation that takes the "infinte roots" in the $x$-coordinate to finite roots, we prove that the Smith normal form stays intact through the transformation, allowing us to lift the earlier assumption.
[3M96] M. G. Marinari, H. M. Möller, T. Mora. On multiplicities in polynomial systems solving. Trans. Amer. Math. Soc., 348(8):3283--3321, 1996.