Authors:V. M. Buchstaber; V. Dragović Abstract: A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of \(\sigma \) -functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were ”virtual”, while in the present case the resulting bundles are effectively realizable. PubDate: 2018-04-09 DOI: 10.1007/s40598-018-0085-2

Authors:Filip D. Jevtić; Marija Jelić; Rade T. Živaljević Abstract: We show that the cyclohedron (Bott–Taubes polytope) \(W_n\) arises as the polar dual of a Kantorovich–Rubinstein polytope \(KR(\rho )\) , where \(\rho \) is an explicitly described quasi-metric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron \(\Delta _{{\widehat{\mathcal {F}}}}\) (associated to a building set \({\widehat{\mathcal {F}}}\) ) and its non-simple deformation \(\Delta _{\mathcal {F}}\) , where \(\mathcal {F}\) is an irredundant or tight basis of \({\widehat{\mathcal {F}}}\) (Definition 21). Among the consequences are a new proof of a recent result of Gordon and Petrov (Arnold Math. J. 3(2):205–218, 2017) about f-vectors of generic Kantorovich–Rubinstein polytopes and an extension of a theorem of Gelfand, Graev, and Postnikov, about triangulations of the type A, positive root polytopes. PubDate: 2018-04-09 DOI: 10.1007/s40598-018-0083-4

Authors:Anton Leykin; Jose Israel Rodriguez; Frank Sottile Abstract: The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space. PubDate: 2018-04-05 DOI: 10.1007/s40598-018-0084-3

Authors:Dima Grigoriev; Nicolai Vorobjov Abstract: We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or, alternatively, via the maximal degree of these polynomials. In sparse setting, the bound involves the number of the monomials. PubDate: 2018-04-03 DOI: 10.1007/s40598-018-0086-1

Authors:Ivan Cherednik Abstract: A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of “picking up residues” due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for \(A_1\) in the spherical case. PubDate: 2018-04-03 DOI: 10.1007/s40598-018-0082-5

Authors:F. Pakovich Abstract: Using dynamical methods we give a new proof of the theorem saying that if A, B, X are rational functions of complex variable z of degree at least two such that \(A\circ X=X\circ B\) and \({\mathbb C}(B,X)={\mathbb C}(z)\) , then the Galois closure of the field extension \({\mathbb C}(z)/{\mathbb C}(X)\) has genus zero or one. PubDate: 2018-02-12 DOI: 10.1007/s40598-018-0081-6

Authors:Oleg Karpenkov Abstract: In this small paper we bring together some open problems related to the study of the configuration spaces of tensegrities, i.e. graphs with stresses on edges. These problems were announced in Doray et al. (Discrete Comput Geom 43:436–466, 2010), Karpenkov et al. (ARS Math Contemp 6:305–322, 2013), Karpenkov (The combinatorial geometry of stresses in frameworks. arXiv:1512.02563 [math.MG], 2017), and Karpenkov (Geometric Conditions of Rigidity in Nongeneric settings, 2016) (by F. Doray, J. Schepers, B. Servatius, and the author), for more details we refer to the mentioned articles. PubDate: 2018-02-02 DOI: 10.1007/s40598-018-0080-7

Authors:Sanjay Ramassamy Abstract: For any positive integer q, the sequence of the Euler up/down numbers reduced modulo q was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of q precise conjectures for the minimal period and for the position at which the sequence starts being periodic. When q is a power of 2, a sequence defined by Arnold appears, and we formulate a conjecture for a simple computation of this sequence. PubDate: 2018-01-22 DOI: 10.1007/s40598-018-0079-0

Authors:William Floyd; Gregory Kelsey; Sarah Koch; Russell Lodge; Walter Parry; Kevin M. Pilgrim; Edgar Saenz Pages: 365 - 395 Abstract: We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena. PubDate: 2017-09-01 DOI: 10.1007/s40598-017-0071-0 Issue No:Vol. 3, No. 3 (2017)

Authors:Nikolai Nadirashvili; Serge Vlăduţ Pages: 397 - 421 Abstract: We develop an integral geometry of stationary Euler equations defining some function w on the Grassmannian of affine lines in \(\mathbb {R}^3\) depending on a putative compactly supported solution (v; p) of the system and deduce some linear differential equations for w. We conjecture that \(w=0\) everywhere and prove that this conjecture implies that \(v=0.\) PubDate: 2017-09-01 DOI: 10.1007/s40598-017-0072-z Issue No:Vol. 3, No. 3 (2017)

Authors:Cordian Riener; Nicolai Vorobjov Pages: 423 - 443 Abstract: Fix any real algebraic extension \(\mathbb K\) of the field \(\mathbb Q\) of rationals. Polynomials with coefficients from \(\mathbb K\) in n variables and in n exponential functions are called exponential polynomials over \({\mathbb K}\) . We study zero sets in \({\mathbb R}^n\) of exponential polynomials over \(\mathbb K\) , which we call exponential-algebraic sets. Complements of all exponential-algebraic sets in \({\mathbb R}^n\) form a Zariski-type topology on \({\mathbb R}^n\) . Let \(P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]\) be a polynomial and denote $$\begin{aligned} V:=\{ (x_1, \ldots , x_n) \in {\mathbb R}^n \> P(x_1, \ldots ,x_n,, e^{x_1}, \ldots ,e^{x_n})=0 \}. \end{aligned}$$ The main result of this paper states that, if the real zero set of a polynomial P is irreducible over \(\mathbb K\) and the exponential-algebraic set V has codimension 1, then, under Schanuel’s conjecture over the reals, either V is irreducible (with respect to the Zariski topology) or each of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of P. In the case of a single exponential (i.e., when P is independent of \(U_2, \ldots , U_n\) ) stronger statements are shown which are independent of Schanuel’s conjecture. PubDate: 2017-09-01 DOI: 10.1007/s40598-017-0073-y Issue No:Vol. 3, No. 3 (2017)

Authors:Gleb Nenashev; Boris Shapiro; Michael Shapiro Abstract: The space \(Pol_d\simeq \mathbb {C}P^d\) of all complex-valued binary forms of degree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition \(\mu \vdash d\) . For each such stratum \(S_\mu ,\) we introduce its secant degeneracy index \(\ell _\mu \) which is the minimal number of projectively dependent pairwise distinct points on \(S_\mu \) , i.e., points whose projective span has dimension smaller than \(\ell _\mu -1\) . In what follows, we discuss the secant degeneracy index \(\ell _\mu \) and the secant degeneracy index \(\ell _{{{\bar{\mu }}}}\) of the closure \({{\bar{S}}}_\mu \) . PubDate: 2017-12-06 DOI: 10.1007/s40598-017-0077-7

Authors:Nikita Kalinin; Mikhail Shkolnikov Abstract: The sum (resp. the sum of squares) of the defects in the triangle inequalities for the area one lattice parallelograms in the first quadrant has a surprisingly simple expression. Namely, let \(f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}\) . Then, where the sum runs by all \(a,b,c,d\in {\mathbb {Z}}_{\ge 0}\) such that \(ad-bc=1\) . We present a proof of these formulae and list several directions for the future studies. PubDate: 2017-11-02 DOI: 10.1007/s40598-017-0075-9

Authors:Alexey Basalaev; Atsushi Takahashi; Elisabeth Werner Abstract: This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund–Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold. PubDate: 2017-11-02 DOI: 10.1007/s40598-017-0076-8

Authors:Dmitry Fuchs; Alexandre Kirillov; Sophie Morier-Genoud; Valentin Ovsienko Abstract: We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton’s essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix. PubDate: 2017-08-22 DOI: 10.1007/s40598-017-0074-x

Authors:Pinaki Mondal Abstract: We study two variants of the following question: “Given two finitely generated \(\mathbb {C}\) -subalgebras \(R_1, R_2\) of \(\mathbb {C}[x_1, \ldots , x_n]\) , is their intersection also finitely generated?” We show that the smallest value of n for which there is a counterexample is 2 in the general case, and 3 in the case that \(R_1\) and \(R_2\) are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of \(\mathbb {C}^n\) and to the moment problem on semialgebraic subsets of \(\mathbb {R}^n\) . The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of \(\mathbb {C}^2\) via key forms of valuations centered at infinity. PubDate: 2017-06-06 DOI: 10.1007/s40598-017-0068-8

Authors:Gaiane Panina Abstract: We describe and study an explicit structure of a regular cell complex \(\mathcal {K}(L)\) on the moduli space M(L) of a planar polygonal linkage L. The combinatorics is very much related (but not equal) to the combinatorics of the permutohedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space M is a sphere, the complex \(\mathcal {K}\) is dual to the boundary complex of the permutohedron.The dual complex \(\mathcal {K}^*\) is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron. PubDate: 2017-05-29 DOI: 10.1007/s40598-017-0070-1

Authors:Pavel Etingof Abstract: We explain a proof of the Broué–Malle–Rouquier conjecture on Hecke algebras of complex reflection groups, stating that the Hecke algebra of a finite complex reflection group W is free of rank W over the algebra of parameters, over a field of characteristic zero. This is based on previous work of Losev, Marin– Pfeiffer, and Rains and the author. PubDate: 2017-04-12 DOI: 10.1007/s40598-017-0069-7

Authors:Dami Lee Abstract: In this paper, we will construct an example of a closed Riemann surface X that can be realized as a quotient of a triply periodic polyhedral surface \(\Pi \subset \mathbb {R}^3\) where the Weierstrass points of X coincide with the vertices of \(\Pi .\) First we construct \(\Pi \) by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface. The symmetries of X allow us to construct hyperbolic structures and various translation structures on X that are compatible with its conformal type. The translation structures are the geometric representations of the holomorphic 1-forms of X. Via the basis of 1-forms we find an explicit algebraic description of the surface that suggests the Fermat’s quartic. Moreover the 1-forms allow us to identify the Weierstrass points. PubDate: 2017-04-12 DOI: 10.1007/s40598-017-0067-9

Authors:Aleksandr V. Pukhlikov Abstract: The famous \(4n^2\) -inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is greater than \(4n^2\mu \) , where \(\mu \) is the multiplicity of the singular point. PubDate: 2016-11-30 DOI: 10.1007/s40598-016-0060-8