Polynomial Singularity Research Articles

Maximal and typical topology of real polynomial singularities

Maximal and typical topology of real polynomial singularities

Solving the cohomological equation for locally Hamiltonian flows, part I - local obstructions

We study the cohomological equation Xu=f for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution u when the flow has saddle loops, which has not been systematically studied before. Then we need to limit the flow to its minimal components. We show the existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs). Our main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni's distributions (cf. [9,11]), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of f around the saddles. Our main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a. locally Hamiltonian flows (with or without saddle loops) to be shown in [15].The main contribution of this article is to define the aforementioned new families of invariant distributions dσ,jk, Cσ,lk and analyze their effect on the regularity of u and on the regularity of the associated cohomological equations for IETs. To prove this new phenomenon, we further develop local analysis of f near degenerate singularities inspired by tools from [14] and [17]. We develop new tools of handling functions whose higher derivatives have polynomial singularities over IETs.

New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

Abstract We consider smooth locally Hamiltonian flows on compact surfaces of genus g ≥ 1 {g\geq 1} to prove a deviation formula of Birkhoff integrals for smooth observables. Our work generalizes results of [G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 2002, 1, 1–103] and [A. I. Bufetov, Limit theorems for translation flows, Ann. of Math. (2) 179 2014, 2, 431–499] (and a recent result in [K. Frączek and C. Ulcigrai, On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions, preprint 2021]) which prove the existence of the deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals display a power deviation spectrum with exactly g positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich–Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the deviation spectrum of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in the deviation spectrum related to the non-vanishing of some derivatives of the observable at fixed points. The proof of this new phenomenon is inspired by tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of interval exchange transformations.

The Probabilistic Method in Real Singularity Theory

We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.

Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets

The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $\partial S=\partial S_0$ and ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $\boldsymbol{Z}$ coefficients, but also for normal currents having $\boldsymbol{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $B\subset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.

Landau–Ginzburg mirror symmetry conjecture

We prove the Landau–Ginzburg mirror symmetry conjecture between invertible quasihomogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito–Givental theory (LG B-model) of the mirror polynomial.

Real algebraic links in S3 and braid group actions on the set of n-adic integers

We construct an infinite tower of covering spaces over the configuration space of [Formula: see text] distinct nonzero points in the complex plane. This results in an action of the braid group [Formula: see text] on the set of [Formula: see text]-adic integers [Formula: see text] for all natural numbers [Formula: see text]. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid [Formula: see text] an infinite sequence of braids, whose braid types are invariants of [Formula: see text]. We present computations for the cases of [Formula: see text] and [Formula: see text] and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials [Formula: see text].

Singularities of base polynomials and Gau–Wu numbers

In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n×n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range W(A). We refer to k(A) as the Gau–Wu number of the matrix A. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu [14] classifying the Gau–Wu numbers for 3×3 matrices. Our focus on singularities is inspired by Chien and Nakazato [3], who classified W(A) for 4×4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n×n matrix, we give necessary conditions for k(A)=2, characterize k(A)=n, and apply these results to the case of unitarily irreducible 4×4 matrices. However, we show that knowledge of the singularities is not sufficient to determine k(A) by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different k(A). In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible A with reducible base curve and show that we can find corresponding matrices with identical base curve but different k(A). Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky [5], generalizing their result in the process.

SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO IMPROVED BOUNDS

Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_{n}$ is singular with probability at most $(2+o(1))^{-n}$ . On the other hand, the best known upper bound on the singularity probability of $M_{n}$ , due to Vershynin (2011), is $2^{-n^{c}}$ , for some unspecified small constant $c>0$ . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_{n}$ is at most $2^{-n^{1/4}\sqrt{\log n}/1000}$ for all sufficiently large $n$ . The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.

κ -Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight

A natural star product for 4-d $\kappa$-Minkowski space is used to investigate various classes of $\kappa$-Poincar\'e invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual $\phi^4$ theory. $\kappa$-Poincar\'e invariance forces the integral involved in the actions to be a twisted trace, thus defining a KMS weight for the noncommutative (C*-)algebra modeling the $\kappa$-Minkowski space. The associated modular group and Tomita modular operator are characterized. In all the field theories, the twist generates different planar one-loop contributions to the 2-point function which are at most UV linearly diverging. Some of these theories are free of UV/IR mixing. In the others, UV/IR mixing shows up in non-planar contributions to the 2-point function as a polynomial singularity at exceptional zero external momenta while staying finite at non-zero external momenta. These results are discussed together with the possibility for the KMS weight relative to the quantum space algebra to trigger the appearance of KMS state on the algebra of observables.

Moduli algebras of some non-semiquasihomogeneous singularities

Under some additional restrictions we find dimensions and bases of moduli algebras of isolated singularities of polynomials in n variables that are sums of n monomials of equal weighted degrees and one monomial of lower degree.

Invertibility of symmetric random matrices

AbstractWe study symmetric random matrices H, possibly discrete, with iid above‐diagonal entries. We show that H is singular with probability at most , and . Furthermore, the spectrum of H is delocalized on the optimal scale . These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdös, Schlein and Yau.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 135‐182, 2014

Constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem with continuous time

Estimates for the rate of convergence in ergodic theorems are necessarily spectral. We find the equivalence constants relating the polynomial rate of convergence in von Neumann’s mean ergodic theorem with continuous time and the polynomial singularity at the origin of the spectral measure of the function averaged over the corresponding dynamical system. We also estimate the same rate of convergence with respect to the decrease rate of the correlation function. All results of this article have obvious exact analogs for the stochastic processes stationary in the wide sense.

Relativistic Gravitational Collapse of a Cylindrical Shell of Dust. II: -- Settling Down Boundary Condition --

We numerically study the dynamics of an imploding hollow cylinder composed of dust. Since there is no cylindrical black hole in 4-dimensional spacetime with physically reasonable energy conditions, a collapsed dust cylinder involves a naked singularity accompanied by its causal future, or a fatal singularity which terminates the history of the whole universe. In a previous paper, the present authors have shown that if the dust is assumed to be composed of collisionless particles such that these particles go through the symmetry axis of the cylinder, then the scalar polynomial singularity formed on the symmetry axis is so weak that almost all of geodesics are complete, and thus effectively no singularity forms by the collapse of a hollow dust cylinder. By contrast, in this paper, we assume that whole of the collapsed dust settles down on the symmetry axis by changing its equation of state. Obtained solutions are the straightforward extension of Morgan's null dust solution, in which no gravitational radiation is emitted. However, in the present case with timelike dust, infinite amount of $C$-energy initially stored in the system is released through gravitational radiation. We also show that the gravitational waves asymptotically behave in a self-similar manner.

Scalar polynomial singularities in power-law spacetimes

Recently, Helliwell and Konkowski have examined the quantum “healing” of some classical singularities in certain power-law spacetimes. Here I further examine classical properties of these spacetimes and show that some of them contain naked strong curvature singularities.

APN monomials over GF( [formula omitted]) for infinitely many n

I present some results towards a complete classification of monomials that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over F 2 n for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, in: Actualités Sci. Ind., vol. 1041, Hermann, Paris, 1948] on absolutely irreducible curves shows that a monomial x m is not APN over F 2 n for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over F 2 . I will show that the latter polynomial's singularities imply that except in three specific, narrowly defined cases, all monomials have such a factor over a finite field of characteristic 2. Two of these cases, those with exponents of the form 2 k + 1 or 4 k − 2 k + 1 for any integer k, are already known to be APN for infinitely many fields. The last, relatively rare case when a certain gcd is maximal is still unproven; my method fails. Some specific, special cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.

Theory of singular perturbations with a non-smooth spectrum of the limit operator

This work is devoted to the development of the regularization method [1] for the case where the spectrum of the limit variable operator has non-smooth singularities. The case will be investigated where one of the eigenvalues of the limit operator vanishes identically on certain sections of the segment , and the approach to these sections has a polynomial singularity. Such problems arise in the radio and impulse technologies.

Strengths of naked singularities in radiation collapse with nonlinear mass functions

The structure of naked singularity for a nonlinear mass function in Vaidya solutions for collapse of imploding radiation is examined here. The singularity is shown to be a scalar polynomial singularity even though RabKaKb does not diverge sufficiently fast along any singular geodesic.

Singularities on the boundary of the hyperbolicity region

The singularities of hyperbolic polynomials (hypersurfaces) and the singularities of the boundary of the hyperbolicity region are investigated. Theorems on stabilization of these singularities in families with a fixed number of parameters and on their relationship with elliptic singularities are proved. The problems considered in this study are part of a research program focusing on singularities of boundaries of spaces of differential equations, proposed by V. I. Arnol'd.

Singularities in separable metrics with spherical, plane, and hyperbolic symmetry

We obtain all possible solutions to the Einstein equations for a perfect fluid with a metric of the form $$ds^2 = - w^2 (x)v^2 (t)dt^2 + g^2 (t)S^2 (x)dx^2 + A^2 (t)B^2 (x)[d$$ which obey the weak and strong energy conditions and do not contain scalar polynomial singularities on surfacesx = const. We show that the only nonstatic solutions satisfying these conditions are the Robertson-Walker spacetimes, spacetimes withw = S = B = 1, and a class of plane symmetric solutions.