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Abstract: Abstract To discuss the validity of the Kummer–Vandiver conjecture, we consider a generalized problem associated to the conjecture. Let p be an odd prime number and \(\zeta _p\) a primitive p-th root of unity. Using new programs, we compute the Iwasawa invariants of \({\textbf{Q}}(\sqrt{d},\zeta _p)\) in the range \( d <200\) and \(200<p <1{,}000{,}000\) . From our data, the actual numbers of exceptional cases seem to be near the expected numbers for \(p<1{,}000{,}000\) . Moreover, we find a few rare exceptional cases for \( d <10\) and \(p>1{,}000{,}000\) . We give two partial reasons why it is difficult to find exceptional cases for \(d=1\) including counter-examples to the Kummer–Vandiver conjecture. PubDate: 2022-11-07
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Abstract: Abstract We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of A-discriminant amoeba complements, and computational real algebraic geometry. PubDate: 2022-10-27
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Abstract: Abstract Let \(\lambda \) be a partition of an integer n and \({\mathbb F}_q\) be a finite field of order q. Let \(P_\lambda (q)\) be the number of strictly upper triangular \(n\times n\) matrices of the Jordan type \(\lambda \) . It is known that the polynomial \(P_\lambda \) has a tendency to be divisible by high powers of q and \(Q=q-1\) , and we put \(P_\lambda (q)=q^{d(\lambda )}Q^{e(\lambda )}R_\lambda (q)\) , where \(R_\lambda (0)\ne 0\) and \(R_\lambda (1)\ne 0\) . In this article, we study the polynomials \(P_\lambda (q)\) and \(R_\lambda (q)\) . Our main results: an explicit formula for \(d(\lambda )\) (an explicit formula for \(e(\lambda )\) is known, see Proposition 3.3 below), a recursive formula for \(R_\lambda (q)\) (a similar formula for \(P_\lambda (q)\) is known, see Proposition 3.1 below), the stabilization of \(R_\lambda \) with respect to extending \(\lambda \) by adding strings of 1’s, and an explicit formula for the limit series \(R_{\lambda 1^\infty }\) . Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields. PubDate: 2022-10-13
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Abstract: Abstract We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself. PubDate: 2022-10-01
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Abstract: Abstract The fundamental theorem of affine geometry says that if a self-bijection f of an affine space of dimenion n over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then f of the expected type, namely f is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type. PubDate: 2022-10-01
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Abstract: Abstract We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or \(\tau \) -model), or as three permutations on the set of half-edges (rotation system or \(\sigma \) -model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality. PubDate: 2022-10-01
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Abstract: Abstract We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform. PubDate: 2022-10-01
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Abstract: Abstract This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021. PubDate: 2022-10-01
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Abstract: Abstract The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let A(t) be an \(n \times n\) matrix whose entries are Laurent series in t. We show that, as \(t \rightarrow 0\) , the logarithms of singular values of A(t) approach the invariant factors of A(t). This leads us to suggest logarithms of singular values of an \(n \times n\) complex matrix as an analog of the logarithm map on \((\mathbb {C}^*)^n\) for the matrix group \({\text {GL}}(n, \mathbb {C})\) . PubDate: 2022-09-16 DOI: 10.1007/s40598-022-00217-y
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Abstract: Abstract The Kepler orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits \(\mathrm { PSL}_2(\mathbb {R})\) as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre \(y^{\prime \prime }= \omega (x, y, y^{\prime })\) , vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new. PubDate: 2022-09-12 DOI: 10.1007/s40598-022-00213-2
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Abstract: Abstract Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in \({\mathbb {C}}^2\) , and we give an asymptotic counting result using lattice point counting techniques. PubDate: 2022-08-29 DOI: 10.1007/s40598-022-00216-z
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Abstract: Abstract We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, such as non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics. PubDate: 2022-08-24 DOI: 10.1007/s40598-022-00215-0
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Abstract: Abstract We describe a topological relationship between slices of the parameter space of cubic maps. In the paper [9], Milnor defined the curves \(\mathcal {S}_{p}\) as the set of all cubic polynomials with a marked critical point of period p. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves \(\mathcal {S}_{1}\) and \(\mathcal {S}_{2}\) . PubDate: 2022-07-22 DOI: 10.1007/s40598-022-00211-4
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Abstract: Abstract We study the categorical entropy and counterexamples to Gromov–Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan–Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston’s classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy. PubDate: 2022-07-18 DOI: 10.1007/s40598-022-00210-5
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Abstract: Abstract We show that there are infinitely many nonisomorphic quandle structures on any topogical space X of positive dimension. In particular, we disprove Conjecture 5.2 in Cheng et al. (Topology Appl 248:64–74, 2018), asserting that there are no nontrivial quandle structures on the closed unit interval [0, 1]. PubDate: 2022-07-12 DOI: 10.1007/s40598-022-00212-3
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Abstract: Abstract We show how the formalism of 2-traces can be applied in the setting of derived algebraic geometry to obtain a generalization of the holomorphic Atiyah–Bott formula to the case when an endomorphism is replaced by a correspondence. PubDate: 2022-07-05 DOI: 10.1007/s40598-022-00206-1
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Abstract: Abstract We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time. PubDate: 2022-06-27 DOI: 10.1007/s40598-022-00208-z
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Abstract: Abstract With any surjective rational map \(f: \mathbb {P}^n \dashrightarrow \mathbb {P}^n\) of the projective space, we associate a numerical invariant (ML degree) and compute it in terms of a naturally defined vector bundle \(E_f \longrightarrow \mathbb {P}^n\) . PubDate: 2022-05-25 DOI: 10.1007/s40598-022-00207-0
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