arXiv subjects

Martin Helmer

Publications and source records attributed to Martin Helmer.

At least 19 recordsLinked to original sources

arXiv2026

Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis

Suppose that $f(x) \in \mathbb{R}[x_1,\dots, x_n]$ and $g(x) \in \mathbb{R}[x_1,\dots, x_n]$ are two real polynomials of degree $d$ in $n$ variables. If the polynomials $f$ and $g$ are the same up to orthogonal symmetry a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element $R\in O(n)$ such that $f(Rx)=g(x)$. One may directly solve this problem by constructing a nonlinear system of equations induced by the relation $f(Rx)=g(x)$ along with the identities of the orthogonal group however this approach becomes quite computationally expensive for larger values of $n$ and $d$. To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analysis (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the $R\in O(n)$ such that $f(Rx)=g(x)$.

math.RA↗

arXiv2025

Geometric Singularities of Feynman Integrals

We provide a new method to calculate the full microlocal description of singularities of Feynman integrals. This is done by associating a unique constructible function to the system of partial differential equations (PDEs) annihilating the integral and from this function the singularities can directly be read-off. This function can be constructed explicitly even if the system of PDEs is unknown and describes both the location of the singularities and the number of master integrals on them. Our framework is flexible enough to preform the calculation in any of the Lee-Pomeransky, Feynman, or momentum representations.

hep-th↗

arXiv2025

Spectral Decomposition of Euler-Mellin Integrals

We consider the spectral decomposition of singularities of integrals and their integrands. Our results apply to any integral of Euler-Mellin type, and thus especially to every scalar Feynman integral. Specifically we provide for both the integrand and integral respectively; two explicit constructions of the characteristic variety and characteristic cycle of the constructible function and $D$-module they are associated with. From this we also obtain the singular locus or Landau singularities of the integral. En route we give a simple procedure to compute the local Euler obstruction function of a variety, and using this, to compute the Euler characteristic of the complex link of a Whitney stratum.

math-ph↗

arXiv2024

Stratification of Projection Maps From Toric Varieties

We prove a combinatorial version of Thom's Isotopy Lemma for projection maps applied to any complex or real toric variety. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all examples we produced, our resulting algorithm outperforms known general purpose methods in Helmer and Nanda (FoCM, 2022), and Dinh and Jelonek (DCG, 2021) for computing map-stratifications.

math.AG↗

arXiv2024

Faster computation of Whitney stratifications and their minimization

We describe two new algorithms for the computation of Whitney stratifications of real and complex algebraic varieties. The first algorithm is a modification of the algorithm of Helmer and Nanda (HN), but is made more efficient by using techniques for equidimensional decomposition rather than computing the set of associated primes of a polynomial ideal at a key step in the HN algorithm. We note that this modified algorithm may fail to produce a minimal Whitney stratification even when the HN algorithm would produce a minimal stratification. The second algorithm coarsens a given Whitney stratification of a complex variety to the unique minimal Whitney stratification; we refer to this as the minimization of a stratification. The theoretical basis for our approach is a classical result of Teissier. To our knowledge this yields the first algorithm for computing a minimal Whitney stratification.

math.AG↗

arXiv2024

Landau Singularities from Whitney Stratifications

We demonstrate that the complete and non-redundant set of Landau singularities of Feynman integrals may be explicitly obtained from the Whitney stratification of a certain map. As a proof of concept, we leverage recent theoretical and algorithmic advances in their computation in order to determine this set for nontrivial examples of two-loop integrals. Interestingly, different strata of the Whitney stratification describe not only the singularities of a given integral, but also those of integrals obtained from kinematic limits, e.g. by setting some of its masses or momenta to zero.

hep-th↗

arXiv2023

Effective Whitney Stratification of Real Algebraic Varieties

We describe new algorithms to compute Whitney stratifications of real algebraic varieties. Using either conormal or polar techniques, these algorithms stratify a complexification of a given real variety. We then show that the resulting stratification can be described by real polynomials. We also extend these methods to stratification problems involving the so-called full semialgebraic sets as well as real algebraic maps.

math.AG↗

arXiv2023

Symbol Alphabets from the Landau Singular Locus

We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal $A$-determinant. Focusing on one loop, we further show that all square-root letters may also be obtained, by re-factorizing the principal $A$-determinant with the help of Jacobi identities. We verify our findings by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta, also finding agreement with earlier results in the literature for the latter case. We provide a computer implementation of our results for the principal $A$-determinants, symbol alphabets and canonical differential equations in an accompanying Mathematica file. Finally, we study the question of when a one-loop integral satisfies the Cohen-Macaulay property and show that for almost all choices of kinematics the Cohen-Macaulay property holds. Throughout, in our approach to Feynman integrals, we make extensive use of the Gel'fand, Graev, Kapranov and Zelevinski\u{\i} theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal $A$-determinant.

hep-th↗

arXiv2021

Segre-Driven Radicality Testing

We present a probabilistic algorithm to test if a homogeneous polynomial ideal $I$ defining a scheme $X$ in $\mathbb{P}^n$ is radical using Segre classes and other geometric notions from intersection theory. Its worst case complexity depends on the geometry of $X$. If the scheme $X$ has reduced isolated primary components and no embedded components supported the singular locus of $X_{\rm red}=V(\sqrt{I})$, then the worst case complexity is doubly exponential in $n$; in all the other cases the complexity is singly exponential. The realm of the ideals for which our radical testing procedure requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly exponential complexity for the ideal membership problem.

math.AG↗

arXiv2021

Cohen-Macaulay Property of Feynman Integrals

The connection between Feynman integrals and GKZ $A$-hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new possibilities; in this paper we continue to explore this connection. To each such hypergeometric system there is an associated toric ideal, we prove that the latter has the Cohen-Macaulay property for two large families of Feynman integrals. This implies, for example, that both the number of independent solutions and dynamical singularities are independent of space-time dimension and generalized propagator powers. Furthermore, in particular, it means that the process of finding a series representation of these integrals is fully algorithmic.

hep-th↗

arXiv2021

Conormal Spaces and Whitney Stratifications

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.

math.AG↗

arXiv2020

Complex Links and Hilbert-Samuel Multiplicities

We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce $X$ to a point $p$ and $Y$ to a curve $C$. Next, we establish that $e_pC$ equals the Euler characteristic (and hence, the cardinality) of the complex link of $p$ in $C$. Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of $p$ in $C$) to determine this Euler characteristic with high confidence.

math.AG↗

arXiv2020

On the Uniqueness Problem for Quadrature Domains

We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic system, which is analyzed using the method of real comprehensive triangular decomposition.

math.CV↗

arXiv2019

Probabilistic Saturations and Alt's Problem

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt's problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

math.AG↗

arXiv2019

Families of Toric Chemical Reaction Networks

We study families of chemical reaction networks whose positive steady states are toric, and therefore can be parameterized by monomials. Families are constructed algorithmically from a core network; we show that if a family member is multistationary, then so are all subsequent networks in the family. Further, we address the questions of model selection and experimental design for families by investigating the algebraic dependencies of the chemical concentrations using matroids. Given a family with toric steady states and a constant number of conservation relations, we construct a matroid that encodes important information regarding the steady state behaviour of the entire family. Among other things, this gives necessary conditions for the distinguishability of families of reaction networks with respect to a data set of measured chemical concentrations. We illustrate our results using multi-site phosphorylation networks.

q-bio.MN↗

arXiv2018

Multistationarity and Bistability for Fewnomial Chemical Reaction Networks

Bistability and multistationarity are properties of reaction networks linked to switch-like responses and connected to cell memory and cell decision making. Determining whether and when a network exhibits bistability is a hard and open mathematical problem. One successful strategy consists of analyzing small networks and deducing that some of the properties are preserved upon passage to the full network. Motivated by this we study chemical reaction networks with few chemical complexes. Under mass-action kinetics the steady states of these networks are described by fewnomial systems, that is polynomial systems having few distinct monomials. Such systems of polynomials are often studied in real algebraic geometry by the use of Gale dual systems. Using this Gale duality we give precise conditions in terms of the reaction rate constants for the number and stability of the steady states of families of reaction networks with one non-flow reaction.

math.AG↗

arXiv2018

Segre Class Computation and Practical Applications

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

math.AG↗

arXiv2017

Polar degrees and closest points in codimension two

Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix $A$ in the codimension two case.

math.AG↗