Classical Algebraic Geometry

Algebraic geometry studies properties of specific algebraic varieties, on the one hand, and moduli spaces of all varieties of fixed topological type on the other hand. Of special importance is the moduli space of curves, whose properties are subject of ongoing research. The rationality versus general type question of these and related spaces is of classical and also very modern interest with recent progress presented in the conference. Certain different birational models of the moduli space of curves and maps have an interpretation as moduli spaces of singular curves and maps. For specific varieties a wide range of questions was addressed, including extrinsic questions (syzygies, the k-secant lemma) and intrinsic ones (generalization of notions of positivity of line bundles, closure operations on ideals and sheaves)

On the bit complexity of polynomial system solving

We describe and analyze a randomized algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence outside a given hypersurface. We show that its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. The algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a “lucky” prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Homotopy techniques for solving sparse column support determinantal polynomial systems

Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F} = [f_{i,j}] \in \mathbf{K}[x_1, \ldots, x_n]^{p \times q}$, with $p \leq q$, we are interested in determining the isolated points of $V_p(\mathbf{F},\mathbf{g})$, the algebraic set of points in $\overline{\mathbf{K}}$ at which all polynomials in $\mathbf{g}$ and all $p$-minors of $\mathbf{F}$ vanish, under the assumption $n = q - p + s + 1$. Such polynomial systems arise in a variety of applications including for example polynomial optimization and computational geometry. We design a randomized sparse homotopy algorithm for computing the isolated points in $V_p(\mathbf{F},\mathbf{g})$ which takes advantage of the determinantal structure of the system defining $V_p(\mathbf{F}, \mathbf{g})$. Its complexity is polynomial in the maximum number of isolated solutions to such systems sharing the same sparsity pattern and in some combinatorial quantities attached to the structure of such systems. It is the first algorithm which takes advantage both on the determinantal structure and sparsity of input polynomials. We also derive complexity bounds for the particular but important case where $\mathbf{g}$ and the columns of $\mathbf{F}$ satisfy weighted degree constraints. Such systems arise naturally in the computation of critical points of maps restricted to algebraic sets when both are invariant by the action of the symmetric group

Computing critical points for invariant algebraic systems

Let $\mathbf{K}$ be a field and $\phi$, $\mathbf{f} = (f_1, \ldots, f_s)$ in $\mathbf{K}[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\mathcal{S}_n$, the group of permutations of $\{1, \dots, n\}$. We consider the problem of computing the points at which $\mathbf{f}$ vanish and the Jacobian matrix associated to $\mathbf{f}, \phi$ is rank deficient provided that this set is finite. We exploit the invariance properties of the input to split the solution space according to the orbits of $\mathcal{S}_n$. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in $d^s$, ${{n+d}\choose{d}}$ and $\binom{n}{s+1}$ where $d$ is the maximum degree of the input polynomials. When $d,s$ are fixed, this is polynomial in $n$ while when $s$ is fixed and $d \simeq n$ this yields an exponential speed-up with respect to the usual polynomial system solving algorithms

Trends and Developments in Complex Dynamics

[no abstract available