Showing papers on "Generic polynomial published in 2019"

Maximal and Typical Topology of Real Polynomial Singularities

Abstract: Given a polynomial map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $\psi$ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a "singularity"). Concerning the upper estimate on the topological complexity of a polynomial singularity, we sharpen the classical bound $b(Z)\leq O(d^{m+1})$, proved by Milnor, with \begin{equation}\label{eq:abstract} b(Z)\leq O(d^{m}),\end{equation} which holds for the generic polynomial map. For what concerns the "lower bound" on the topology of $Z$, we prove a general semicontinuity result for the Betti numbers of the zero set of $\mathcal{C}^0$ perturbations of smooth maps -- the case of $\mathcal{C}^1$ perturbations is the content of Thom's Isotopy Lemma (essentially the Implicit Function Theorem). This result is of independent interest and it is stated for general maps (not just polynomial); this result implies that small continuous perturbations of $\mathcal{C}^1$ manifolds have a richer topology than the one of the original manifold. We then compare the extremal case with a random one and prove that on average the topology of $Z$ behaves as the "square root" of its upper bound: for a random Kostlan map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we have: \begin{equation} \mathbb{E}b(Z)=\Theta(d^{\frac{m}{2}}).\end{equation} This generalizes classical results of Edelman-Kostlan-Shub-Smale from the zero set of a random map, to the structure of its singularities.

On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

Abstract: We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.

A note on generic polynomial maps having a fiber of maximal dimension

Abstract: For any two integers $k,n$, $2\leq k\leq n$, let $f:(\mathbb{C}^*)^n\rightarrow\mathbb{C}^k$ be a generic polynomial map with given Newton polytopes. It is known that points, whose fiber under $f$ has codimension one, form a finite set $C_1(f)$ in $\mathbb{C}^k$. For maps $f$ above, we show that $C_1(f)$ is empty if $k\geq 3$, we classify all Newton polytopes contributing to $C_1(f) eq \emptyset$ for $k=2$, and we compute $|C_1(f)|$.

A note on polynomial maps having fibers of maximal dimension

Abstract: For any two integers $k,n$, $2\leq k\leq n$, let $f:(\mathbb{C}^*)^n\rightarrow\mathbb{C}^k$ be a generic polynomial map with given Newton polytopes. It is known that points, whose fiber under $f$ has codimension one, form a finite set $C_1(f)$ in $\mathbb{C}^k$. For maps $f$ above, we show that $C_1(f)$ is empty if $k\geq 3$, we classify all Newton polytopes contributing to $C_1(f) eq \emptyset$ for $k=2$, and we compute $|C_1(f)|$.

The $p$-adic Riemann Hypothesis For Expnonential Sums.

Abstract: The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is defined. A $p$-adic Riemann hypothesis is formulated. It asserts that the Newton polygon of the $L$-function coincides with the Frobenius polygon when $p$ is large enough. This $p$-adic Riemann hypothesis is proved when $n=2$ and $p\equiv-1({\rm mod }\ d)$. In general, it is proved that the Newton polygon of the $L$-function lies above the Frobenius polygon with coincide endpoints when $p$ is large enough.

Expnonential Sums associated to the Generic polynomial

Abstract: The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is defined. A $p$-adic Riemann hypothesis is formulated. It asserts that the Newton polygon of the $L$-function coincides with the Frobenius polygon when $p$ is large enough. This $p$-adic Riemann hypothesis is proved when $n=2$ and $p\equiv-1({\rm mod }\ d)$. In general, it is proved that the Newton polygon of the $L$-function lies above the Frobenius polygon with coincide endpoints when $p$ is large enough.