About a non-standard interpolation problem

Using algebraic methods, and motivated by the one variable case, we study a multipoint interpolation problem in the setting of several complex variables. The duality realized by the residue generator associated with an underlying Gorenstein algebra, using the Lagrange interpolation polynomial, plays a key role in the arguments

La géometrie du lieu flex d'une hypersurface

International audienceWe give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces in P3. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface

Sub-cubic Change of Ordering for Gröner Basis: A Probabilistic Approach

International audienceThe usual algorithm to solve polynomial systems using Gröbner bases consists of two steps: first computing the DRL Gröbner basis using the F5 algorithm then computing the LEX Gröbner basis using a change of ordering algorithm. When the Bézout bound is reached, the bottleneck of the total solving process is the change of ordering step. For 20 years, thanks to the FGLM algorithm the complexity of change of ordering is known to be cubic in the number of solutions of the system to solve. We show that, in the generic case or up to a generic linear change of variables, the multiplicative structure of the quotient ring can be computed with no arithmetic operation. Moreover, given this multiplicative structure we propose a change of ordering algorithm for Shape Position ideals whose complexity is polynomial in the number of solutions with exponent ω where 2 ≤ ω < 2.3727 is the exponent in the complexity of multiplying two dense matrices. As a consequence, we propose a new Las Vegas algorithm for solving polynomial systems with a finite number of solutions by using Gröbner basis for which the change of ordering step has a sub-cubic (i.e. with exponent ω) complexity and whose total complexity is dominated by the complexity of the F5 algorithm. In practice we obtain significant speedups for various polynomial systems by a factor up to 1500 for specific cases and we are now able to tackle some instances that were intractable