arXiv subjects

László M. Fehér

Publications and source records attributed to László M. Fehér.

10 recordsLinked to original sources

arXiv2026

Thom series in negative relative codimension

We develop a theory of Thom series for contact function singularities. Quadratic stabilization $σ_q$ of a contact singularity $η\in J^k(n,1)$ gives contact singularities $σ_q^{i}η\in J^k(n+i,1)$. We study the stable Thom polynomials (i.e.\ the Thom polynomial in quotient variables) of $σ_q^{i}η$ as $i$ increases. To this end, we define the Thom series of a contact function singularity $η$, which for any given $i$ is a linear combination of Schur polynomials $s_λ$. For large enough $i$ the value of the Thom series at $1-(n+i)$ is the stable Thom polynomial of $σ_q^iη$ and for small $i$ it determines its coefficients outside the kernel of a specialization map. We prove that the Thom series has two main properties: 1) as $i$ increases, the partitions $λ$ follow a simple stabilization pattern, and there is a finite set of $λ$ which generates the support of the entire Thom series via this stabilization; 2) the coefficients of the $s_λ$ are polynomials in $i$ with explicit degree bounds. We describe the precise relationship between unstable and stable Thom polynomials of contact function singularities and Legendre Thom polynomials and we carry out computations of all three, as follows. We compute the complete family of stable Thom polynomials for function singularities with $γ\leq 6$. We compute unstable and Legendre Thom polynomials for several families of binary and ternary singularities. We define a class of multi-binary singularities, and compute their Thom polynomials. We discuss second order Thom-Boardman classes, and indicate difficulties that arise beyond function singularities. The results of the paper will be used in a companion paper, where Thom polynomials will be applied to problems in enumerative geometry.

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arXiv2026

Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics

We study the symmetric polynomial $\prod_{α\in A_{n,d}}\bigl(1+α_1 x_1+\cdots+α_n x_n\bigr)$ where $A_{n,d}:=\{α\in\mathbb{Z}_{\ge 0}^n:|α|=d\}$, which is the total Chern class of $\mathrm{Sym}^d(\mathbb{C}^n)$, viewed as a torus representation whose Chern roots are the weights $α_1 x_1+\cdots+α_n x_n$ for $α\in A_{n,d}$. Its homogeneous degree-$k$ part $c_k(n,d)$ is the $k$-th Chern class of $\mathrm{Sym}^d(\mathbb{C}^n)$. These Chern classes, together with their coefficients in various symmetric function bases, play a central role in enumerative geometry. Despite their simple definition, general closed formulas for their coefficients are subtle, and many structural properties of these classes have remained poorly understood. In this paper we prove several conjectures concerning their structure, establish explicit formulas, and study log-concavity properties for both the Chern classes and their $K$-theoretic analogue. In rank two, passing to the Schur basis and expanding the Schur coefficients in the binomial basis of $d$, we uncover a new binomial log-concavity phenomenon and prove refined positivity results. The paper demonstrates a novel methodology: we combine several AI systems with human mathematical insight in a coordinated workflow, deploying each tool according to its strengths in experimental discovery, conjecture formation, symbolic proof construction, and verification. To our knowledge, this is one of the first detailed case studies of orchestrating multiple AI tools to make substantial progress on a coherent mathematical research project.

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arXiv2025

Polynomiality of the Striling coefficients of $c(\mathrm{Pol}^d(\mathbb{C}^n))$ and Fano schemes

We prove that the $d$-dependence of $c(\mathrm{Pol}^d(\mathbb{C}^n))$, the Chern class of the $\mathrm{GL}(n)$-representation of degree $d$ homogeneous polynomials in $n$ complex variables is polynomial. We also study the asymptotics of the polynomial $d$-dependence of this Chern class. We apply these results to solve a conjecture of Manivel on the degree of varieties of hypersurfaces containing linear subspaces. We also prove new formulas for the degree and Euler characteristics of Fano schemes of lines for generic hypersurfaces. To prove the polynomial dependence of $c(\mathrm{Pol}^d(\mathbb{C}^n))$ we introduce the notion of Striling coefficients. This can be considered as a generalization of Stirling numbers. We express the Stirling coefficients in terms of specializations of monomial symmetric polynomials so we can deduce their polynomiality and, in certain cases, their asymptotic behaviour.

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arXiv2025

Plücker formulas using equivariant cohomology of coincident root strata

We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Plücker formulas are polynomials in the degree, just as the classical Plücker formulas counting the bitangents and flexes of a degree $d$ generic plane curve. We establish an upper bound for the degrees of these polynomials, and we calculate the leading terms of those whose degrees reach this upper bound. We believe that the paper is understandable without detailed knowledge of equivariant cohomology. It may serve as a demonstration of the use of equivariant cohomology in enumerative geometry through the examples of coincident root strata. We also explain how the equivariant method can be "translated" into the traditional non-equivariant method of resolutions.

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arXiv2025

Obstructions for Morin and fold maps: Stiefel-Whitney classes and Euler characteristics of singularity loci

For a singularity type $η$, let the $η$-avoiding number of an $n$-dimensional manifold $M$ be the lowest $k$ for which there is a map $M\to\mathbb{R}^{n+k}$ without $η$ type singular points. For instance, the case of $η=Σ^1$ is the case of immersions, which has been extensively studied in the case of real projective spaces. In this paper we study the $η$-avoiding number for other singularity types. Our results come in two levels: first we give an abstract reasoning that a non-zero cohomology class is supported on the singularity locus $η(f)$, proving that $η(f)$ cannot be empty. Second, we interpret this obstruction as a non-zero invariant of the singularity locus $η(f)$ for generic $f$. The main technique that we employ is Sullivan's Stiefel-Whitney classes, which are mod 2, real analogues of the Chern-Schwartz-MacPherson (CSM) classes. We introduce the Segre-Stiefel-Whitney classes of a singularity ${\rm s}^{\rm sw}_η$ whose lowest degree term is the mod 2 Thom polynomial of $η$. Using these techniques we compute some universal formulas for the Euler characteristic of a singularity locus.

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arXiv2024

Additive Combinatorics Using Equivariant Cohomology

We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erdős-Heilbronn conjecture. We generalize a theorem of G. Kós (the Grashopper problem) which in some sense is a simultaneous generalization of the Erdős-Heilbronn conjecture. We also prove a signed version of the Erdős-Heilbronn conjecture and the Grashopper problem. Most identities used are based on calculating the projective degree of an algebraic variety in two different ways.

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arXiv2020

Motivic Chern classes of cones

We study motivic Chern classes of cones. First we show examples of projective cones of smooth curves such that their various $K$-classes (sheaf theoretic, push-forward and motivic) are all different. Then we show connections between the torus equivariant motivic Chern class of a projective variety and of its affine cone, generalizing results on projective Thom polynomials.

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arXiv2020

Halving spaces and lower bounds in real enumerative geometry

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $Γ$ with additional cohomological properties. For $Γ=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $Γ=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.

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arXiv2014

Real solutions of a problem in enumerative geometry

We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In the real case the number of solutions can be different on the distinct connected components of the configuration space, resulting in a solution function. The cohomology calculation in the real case only gives the signed sum of the solutions, therefore in general it only gives a lower bound on the range of the solution function. We calculate the solution function for the 2-parameter family and we show that in the even cases the solution function is constant modulo 4. We show how to determine the sign of a solution and describe the connected components of the configuration space. We translate the problem to the language of quivers and also give a geometric interpretation of the sign. Finally, we discuss what aspects might be considered when solving other real enumerative problems.

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arXiv2007

The incidence class and the hierarchy of orbits

R. Rimányi defined the incidence class of two singularities X and Y as $[X]|_Y$, the restriction of the Thom polynomial of X to Y. He conjectured that (under mild conditions) the incidence is not zero if and only if Y is in the closure of X. Generalizing this notion we define the incidence class of two orbits X and Y of a representation. We give a sufficient condition (positivity) for Y to have the property that the incidence class $[X]|_Y$ is not zero if and only if Y is in the closure of X for any other orbit X. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn't hold for all orbits.

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