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Preprints
Preprints
Coauthors: Julianne Barnhart, Michael Burr, Thomas Yahl
Abstract. We relate the monodromy group of a branched cover to the monodromy groups of its restrictions. We show that the presence of a transposition in a restricted monodromy group significantly constrains the monodromy group of the original branched cover. Using this result, we complete the sparse trace test algorithm, which numerically verifies whether a set of solutions to a sparse polynomial system is complete. Along the way, we develop a generalization of the notion of a trace test in the setting of branched covers.
Abstract. We prove there are 1442 four-bar coupler curves through nine generic points in the plane, and thus resolve Alt’s problem. We obtain this proof in three steps. First, we identify the space of coupler curves with a Zariski open subset of Gr(3, 6). Next, we formulate the polynomial system representing the nine-point path synthesis problem in these coordinates and modify it to obtain the mixed volume 5538. Finally, we prove that 4096 of the branches of the generic sparse polynomial system with that support escape the torus in the sparse limit. Thus, we obtain an upper bound of 5538 − 4096 = 1442 for the generic solution count. A lower bound of 1442 is achieved via numerical certification on one instance
Coauthors: Rainer Sinn
Abstract. We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler.
Abstract. We exhibit three conics with 144 tritangent circles, disproving the conjectured maximum of 136.
How to recover a permutation group amidst errors
How to recover a permutation group amidst errors
(B., Kim)
We consider the problem of recovering a permutation group G ≤ S_n from an error-prone sampling process X. We model X as an S_n -valued random variable, defined as a mixture of the uniform distributions on G and S_n . Our suite of tools recovers properties of G from X and bolsters our main method for recovering G itself. Our algorithms are motivated by the numerical computation of monodromy groups, a setting where such error-prone sampling procedures occur organically.
Homotopy Iterators
Homotopy Iterators
(B., Breiding, Friedman)
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which tracks them along the homotopy. Doing so produces a representation of the target solutions, bypassing the need to hold all solutions in memory. We discuss several applications of this datatype ranging from solution counting to data compression.
Accepted/Published
Accepted/Published
Low-Memory Numerical Certification (accepted in ICMS proceedings 2026)
Low-Memory Numerical Certification (accepted in ICMS proceedings 2026)
(B., Breiding, Johnson)
We introduce a low-memory framework for certifying numerical solutions to polynomial systems which uses solution iterators and spatial partitioning trees to reduce memory requirements. We provide a prototypical algorithm, analyze its complexity, and demonstrate the memory reduction on a large example.
Marvelous Slices of Orthogonal Matrices (accepted in Experimental Math)
Marvelous Slices of Orthogonal Matrices (accepted in Experimental Math)
(B., Gesmundo)
The space of 4×4 special orthogonal matrices with zeros on the diagonal decomposes into the union of 14 irreducible surfaces whose intersections are beautifully encoded by the cuboctahedron. Using this decomposition, we exhibit a totally real witness set for SO(4). We explain how to obtain a similar decomposition for SO(5), where the 64 components can be grouped to obtain such a correspondence with the face lattice of a 3-polytope. We show that no such pattern exists for SO(6).
A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schrödinger operators (accepted in Selecta)
A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schrödinger operators (accepted in Selecta)
(B., Faust, Liu)
Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schrödinger operator is Fermi isospectral to the zero potential. This provides a negative answer to a question posed by the third author concerning the rigidity of Fermi isospectrality in dimension two. This example also disproves a conjecture of Gieseker, Knörrer, and Trubowitz in the 1990s stating that for any nontrivial real-valued periodic potential in dimension two, the Fermi variety is irreducible at all energy levels.
Quatroids and Rational Plane Cubics
Quatroids and Rational Plane Cubics
Brysiewicz, T., Gesmundo, F. & Steiner, A. Quatroids and rational plane cubics. Beitr Algebra Geom 65, 923–972 (2024). https://doi.org/10.1007/s13366-024-00776-5
It is a classical result that there are 12 (irreducible) rational cubic curves through 8 generic points in ℙ2ℂ, but little is known about the non-generic cases. The space of 8-point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all 779777 quatroids on eight distinct points in the plane, which produces a full description of the stratification. For each stratum, we generate several invariants, including the number of rational cubics through a generic configuration. As a byproduct of our investigation, we obtain a collection of results regarding the base loci of pencils of cubics and positive certificates for non-rationality.
Lawrence Lifts, Matroids, and Maximum Likelihood Degrees
Lawrence Lifts, Matroids, and Maximum Likelihood Degrees
Accepted to Algebraic Statistics
(B., A. Maraj)
We express the maximum likelihood (ML) degrees of a family toric varieties in termsof M¨obius invariants of matroids. The family of interest are those parametrized by monomial mapsgiven by Lawrence lifts of totally unimodular matrices with even circuits. Specifying these matricesto be vertex-edge incidence matrices of bipartite graphs gives the ML degrees of some hierarchicalmodels and three dimensional quasi-independence models. Included in this list are the no-three-wayinteraction models with one binary random variable, for which, we give closed formulae.
Solving the area-length systems in discrete gravity using homotopy continuation
Solving the area-length systems in discrete gravity using homotopy continuation
Accepted - Classical and Quantum Gravity
Seth K Asante and Taylor Brysiewicz 2024 Class. Quantum Grav. 41 185006
Area variables are intrinsic to connection formulations of general relativity, in contrast to the fun- damental length variables prevalent in metric formulations. Within 4D discrete gravity, particularly based on triangulations, the area-length system establishes a relationship between area variables associated with triangles and the edge length variables. This system is comprised of polynomial equations derived from Heron’s formula, which relates the area of a triangle to its edge lengths.
Using tools from numerical algebraic geometry, we study the area-length systems. In particular, we show that given the ten triangular areas of a single 4-simplex, there could be up to 64 compatible sets of edge lengths. Moreover, we show that these 64 solutions do not, in general, admit formulae in terms of the areas by analyzing the Galois group, or monodromy group, of the problem. We show that by introducing additional symmetry constraints, it is possible to obtain such formulae for the edge lengths. We take the first steps toward applying our results within discrete quantum gravity, specifically for effective spin foam models.
Using tools from numerical algebraic geometry, we study the area-length systems. In particular, we show that given the ten triangular areas of a single 4-simplex, there could be up to 64 compatible sets of edge lengths. Moreover, we show that these 64 solutions do not, in general, admit formulae in terms of the areas by analyzing the Galois group, or monodromy group, of the problem. We show that by introducing additional symmetry constraints, it is possible to obtain such formulae for the edge lengths. We take the first steps toward applying our results within discrete quantum gravity, specifically for effective spin foam models.
Monodromy Coordinates
Monodromy Coordinates
Brysiewicz, T. (2024). Monodromy Coordinates. In: Buzzard, K., Dickenstein, A., Eick, B., Leykin, A., Ren, Y. (eds) Mathematical Software – ICMS 2024. ICMS 2024. Lecture Notes in Computer Science, vol 14749. Springer, Cham. https://doi.org/10.1007/978-3-031-64529-7_28
We introduce the concept of monodromy coordinates for representing solutions to large polynomial systems. Representing solutions this way provides a time-memory trade-off in a monodromy solving algorithm. We describe an algorithm, which interpolates the usual monodromy solving algorithm, for computing such a representation and analyze its space and time complexity.
Max-Convolution through numerics and tropical geometry
Max-Convolution through numerics and tropical geometry
Brysiewicz, T., Hauenstein, J.D. & Hills, C. Max-convolution through numerics and tropical geometry. Numer Algor 96, 845–877 (2024). https://doi.org/10.1007/s11075-023-01668-w
The maximum function, on vectors of real numbers, is not differentiable. Consequently, several differentiable approximations of this function are popular substitutes. We survey three smooth functions which approximate the maximum function and analyze their convergence rates. We interpret these functions through the lens of tropical geometry, where their performance differences are geometrically salient. As an application, we provide an algorithm which computes the max-convolution of two integer vectors in quasi-linear time. We show this algorithm's power in computing adjacent sums within a vector as well as computing service curves in a network analysis application.
The algebraic matroid of the Heron variety
The algebraic matroid of the Heron variety
(S. K. Asante, B., M. Hatzel)
We introduce the n-th Heron variety as the realization space of the (squared) volumes of faces of an n-simplex. Our primary goal is to understand the extent to which Heron's formula, which expresses the area of a triangle as a function of its three edge lengths, can be generalized. Such a formula for one face volume of an n-simplex in terms of other face volumes expresses a dependence in the algebraic matroid of the Heron variety. Whether the volume is expressible in terms of radicals is controlled by the monodromy groups of the coordinate projections of the Heron variety onto coordinates of bases. We discuss a suite of algorithms, some new, for determining these matroids and monodromy groups. We apply these algorithms toward the smaller Heron varieties, organize our findings, and interpret the results in the context of our original motivation.
Code
Polyhedral Geometry in OSCAR
Polyhedral Geometry in OSCAR
Brysiewicz, T., Joswig, M. (2025). Polyhedral Geometry. In: Decker, W., Eder, C., Fieker, C., Horn, M., Joswig, M. (eds) The Computer Algebra System OSCAR. Algorithms and Computation in Mathematics, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-031-62127-7_5
OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Here, we give an introduction to polyhedral geometry computations in OSCAR, as a chapter of the upcoming OSCAR book. In particular, we define polytopes, polyhedra, and polyhedral fans, and we give a brief overview about computing convex hulls and solving linear programs. Three detailed case studies are concerned with face numbers of random polytopes, constructions and properties of Gelfand-Tsetlin polytopes, and secondary polytopes.
Enumerating chambers of hyperplane arrangements with symmetry
Enumerating chambers of hyperplane arrangements with symmetry
Brysiewicz, T., Eble, H. & Kühne, L. Computing Characteristic Polynomials of Hyperplane Arrangements with Symmetries. Discrete Comput Geom 70, 1356–1377 (2023). https://doi.org/10.1007/s00454-023-00557-2
We introduce a new algorithm for enumerating chambers of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase ourjulia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science.
Sparse trace tests
Sparse trace tests
Brysiewicz, T., Burr, M. (2023) Math. Comp. 92 (2023), 2893-2922
DOI: https://doi.org/10.1090/mcom/3849
We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These algorithms extend the classical trace test in numerical algebraic geometry. Our results rely on both the analysis of the structure of sparse resultants as well as an extension of Esterov's results on monodromy groups of sparse systems.
Likelihood degenerations
Likelihood degenerations
(D. Agostini, T. Brysiewicz, C. Fevola, L. Kuhne, B. Sturmfels, S. Telen)
Advances in Mathematics 451 (2022) 108863
Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the variety is smooth, it coincides with the Euler characteristic. We introduce degeneration techniques that are inspired by the soft limits in CEGM theory, and we answer several questions raised in the physics literature. These pertain to bounded regions in discriminantal arrangements and to moduli spaces of point configurations. We present theory and practise, connecting complex geometry, tropical combinatorics, and numerical nonlinear algebra.
Tangent Quadrics in Real 3-Space
Tangent Quadrics in Real 3-Space
(B., C. Fevola, B. Sturmfels)
Le Matematiche 76 (2021) 355-367
We examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of polynomial equations, also in the space of complete quadrics, and we solve them using certified numerical methods. Our aim is to show that Schubert’s problems are fully real
Nodes on quintic spectrahedra
Nodes on quintic spectrahedra
(B., K. Kozhasov, M. Kummer)
Le Matematiche 76 (2021) 415-430
Solving decomposable sparse systems
Solving decomposable sparse systems
(B., J. Rodriguez, F. Sottile, and T. Yahl)
Brysiewicz, Taylor & Rodriguez, Jose & Sottile, Frank & Yahl, Thomas. (2021). Solving decomposable sparse systems. Numerical Algorithms. 10.1007/s11075-020-01045-x.We present a recursive algorithm to solve decomposable sparse systems.
The degree of the Stiefel manifold
The degree of the Stiefel manifold
(B.)
Mathematics in Computer Science (2020)We present an implementation in Macaulay2 of numerical software which computes the Newton polytope of a hypersurface given as the image of a map.We also describe how one can use this as a tropical membership test.
Supplementary Materials
(B.)
J. Symb. Comput. (2019) https://doi.org/10.1016/j.jsc.2019.11.002We count the number of polynomial parametrized osculants to a general analytic curve in the plane: as combinatorial necklaces.
Supplementary Materials
(Brandt, Bruce, B., Krone, Robeva)
Combinatorial Algebraic Geometry, 207-224, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.We compute the degree of SO(n) as an algebraic variety in C^(n^2). We give a combinatorial description of this number as a non-intersecting lattice path count and we relate this degree with the complexity of a low-rank relaxation of semi-definite programming.
Supplementary Materials
(Balay-Wilson, B.)
Rose-Hulman Undergraduate Mathematics Journal. Volume 15, No. 1, Spring 2014.We give an explicit geometric condition for when, for a point on a cubic, there exists another cubic intersecting it exactly at that point (with multiplicity 9).
Supplementary Materials
Dissertation
Dissertation
Newton polytopes and numerical algebraic geometry
Newton polytopes and numerical algebraic geometry
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry.
Supplementary Materials
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