Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
Journal Prestige (SJR): 0.339 Number of Followers: 2 Partially Free Journal ISSN (Print) 01384821  ISSN (Online) 21910383 Published by SpringerVerlag [2468 journals] 
 Reembeddings of affine algebras via Gröbner fans of linear ideals

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Abstract: Abstract Given an affine algebra \(R=K[x_1,\dots ,x_n]/I\) over a field K, where I is an ideal in the polynomial ring \(P=K[x_1,\dots ,x_n]\) , we examine the task of effectively calculating reembeddings of I, i.e., of presentations \(R=P'/I'\) such that \(P'=K[y_1,\dots ,y_m]\) has fewer indeterminates. For cases when the number of indeterminates n is large and Gröbner basis computations are infeasible, we have introduced the method of Zseparating reembeddings in Kreuzer et al. (J Algebra Appl 21, 2022) and Kreuzer, et al. (São Paulo J Math Sci, 2022). This method tries to detect polynomials of a special shape in I which allow us to eliminate the indeterminates in the tuple Z by a simple substitution process. Here we improve this approach by showing that suitable candidate tuples Z can be found using the Gröbner fan of the linear part of I. Then we describe a method to compute the Gröbner fan of a linear ideal, and we improve this computation in the case of binomial linear ideals using a cotangent equivalence relation. Finally, we apply the improved technique in the case of the defining ideals of border basis schemes.
PubDate: 20240131

 Bisector fields of quadrilaterals

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Abstract: Abstract A bisector field is a maximal set \({\mathbb {B}}\) of paired lines in the plane such that each line in each pair crosses the other pairs in \({\mathbb {B}}\) in pairs of points that all share the same midpoint. We showed in a previous article that bisector fields are precisely the sets of pairs of lines that occur as asymptotes of hyperbolas from a pencil of affine conics, along with pairs of parallel lines arising from degenerate parabolas in the pencil. In this article we give a different application, this time to complete quadrilaterals and their ninepoint conics. We show that every complete quadrilateral generates a bisector field as the set of bisectors of the quadrilateral paired according to an orthogonality condition in a geometry determined by the quadrilateral. The ninepoint conic, so named because it passes through nine distinguished points of the quadrilateral, is shown to be the locus of midpoints of the bisector field associated to the quadrilateral, thus giving an interpretation of the other points on the ninepoint conic. Our approach is analytic, and our results hold over any field of characteristic other than 2.
PubDate: 20240131

 On Crofton’s type formulas and the solid angle of convex sets

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Abstract: Abstract Here we analyze three dimensional analogues of the classic Crofton formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in the Euclidean space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has perimeter, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integral by means of integrals of functions related to the solid angle.
PubDate: 20240113

 Algebraic overshear density property

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Abstract: Abstract We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.
PubDate: 20240102

 Archimedean Representation Theorem for modules over a commutative ring

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Abstract: Pólya’s Positivstellensatz and Handelman’s Positivstellensatz are known to be concrete instances of the abstract Archimedean Representation Theorem for (commutative unital) rings. We generalise the Archimedean Representation Theorem to modules over rings. For example, consider the module of all symmetric matrices with entries in a polynomial ring, also known as matrix polynomials. Pólya’s Positivstellensatz and Handelman’s Positivstellensatz had been generalised by Scherer and Hol, and Lê and Du’ respectively to matrix polynomials, using the method of effective estimates from analysis. We show that these two Positivstellensätze for matrix polynomials are concrete instances of our Archimedean Representation Theorem in the case of the module of symmetric matrix polynomials over the polynomial ring.
PubDate: 20231226

 Fundamental Theorem for (A, H)dimodules

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Abstract: Abstract Let k be a field, H a Hopf algebra with a bijective antipode, and A an Hdimodule algebra. We assume that there is an Hcolinear algebra map from H to A. We generalize the Fundamental Theorem of (A, H)Hopf modules to (A, H)dimodules. For example, A could be a coquasitriangular Hopf algebra. When H is the group algebra kG of a multiplicatively abelian group G, an (A, kG)dimodule is just a Ggraded Amodule which is an \(A\#G\) module and a Gdimodule. We also prove the Fundamental Theorem for Hopf Yetter–Drinfeld (A, H)modules, when H is cocommutative and A is a Hopf Yetter–Drinfeld Hmodule algebra. For example, A could be the regular Hopf Yetter–Drinfeld Hmodule algebra H or \(H^{op}\) .
PubDate: 20231224

 Diameter and circumradius of reduced spherical polygons

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Abstract: Abstract In this paper, we show that the reduced spherical regular polygons of any fixed thickness have the minimal diameters and minimal circumradii among all reduced spherical polygons of the same thickness.
PubDate: 20231222

 The real Abelian main conjecture in the finite non semisimple case

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Abstract: Abstract Let \(K/\mathbb {Q}\) be a real cyclic extension of degree divisible by p. We analyze the statement of the “Real Abelian Main Conjecture”, for the pclass group \({\mathscr {H}}_K\) of K. The classical algebraic definition of the padic isotypic components, \({\mathscr {H}}^\textrm{alg}_{K,\varphi }\) , for padic characters \(\varphi = \varphi _0^{} \varphi _p\) ( \(\varphi _0\) of primetop order, \(\varphi _p\) of ppower order), is inappropriate with respect to analytic formulas, because of capitulation of pclasses in the psubextension of \(K/\mathbb {Q}\) . In the 1970’s we have given an arithmetic definition, \({\mathscr {H}}^{\textrm{ar}}_{K,\varphi }\) , and formulated the conjecture, still unproven, \(\#{\mathscr {H}}^{\textrm{ar}}_{K,\varphi } = \#({\mathscr {E}}_K / \widehat{{\mathscr {E}}}_K \, {\mathscr {F}}_{\!K})_{\varphi _{0}^{}}\) , in terms of units \({\mathscr {E}}_K\) , \(\widehat{{\mathscr {E}}}_K\) (units of the strict subfields) and \({\mathscr {F}}_{\!K}\) (Leopoldt’s cyclotomic units). We prove here that the conjecture holds as soon as there exists a prime \(\ell \) , totally inert in K, such that \({\mathscr {H}}_K\) capitulates in \(K(\mu _\ell ^{})\) , existence having been checked, in various circumstances, as a promising new tool. An Appendix of numerical examples is given with PARI programs. A second Appendix deals with the special case \( K \cap \mathbb {Q}(\mu _{p^\infty }^{})^+ \ne \mathbb {Q}\) .
PubDate: 20231212

 Comparable overrings of a commutative ring

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Abstract: Abstract Let \({\mathcal {H}}\) be the set of all commutative rings R such that \(\text {Nil}(R)\) is a divided prime ideal of R and let \(\phi : T(R) \rightarrow R_{\text {Nil}(R)}\) be a ring homomorphism defined as \(\phi (x) = x\) for all \(x \in T(R)\) . An overring \(R_o\) of an integral domain R is said to be comparable if \(R_o\ne R\) , \(R_o \ne \text {qf}(R)\) , and each overring of R is comparable to \(R_o\) under inclusion. We study comparable overrings of a ring in class \({\mathcal {H}}\) .
PubDate: 20231204

 The triangle comparison theorem for spheres

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Abstract: Abstract If two spherical triangles lying on a pair of spheres of different radii have the same edgelengths, then each interior angle of the one on the sphere of larger radius is smaller than the corresponding interior angle of the other one. This is a consequence of Toponogov’s triangle comparison theorem. We prove this fact in an elementary way.
PubDate: 20231201
DOI: 10.1007/s13366022006668

 Rational curves and maximal rank in multiprojective spaces

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Abstract: Abstract We study the multigraded Hilbert function of general rational curves of prescribed multidegree contained in a multiprojective space Y. We have strong negative results, e.g. for almost all line bundles \({\mathcal {L}}\) on Y there is a multidegree such that maximal rank with respect to \({\mathcal {L}}\) fails for all smooth rational curves C of that multidegree, i.e. \(h^0({\mathcal {I}}_C\otimes {\mathcal {L}})>0\) and \(h^1({\mathcal {I}}_C\otimes {\mathcal {L}})>0\) . This is different from the case of general rational curves in projective spaces. We also have positive results, most of them being for the case \(Y =\mathbb {P}^n\times \mathbb {P}^1\) , \(n\ge 2\) .
PubDate: 20231201
DOI: 10.1007/s1336602200661z

 Channel linear Weingarten surfaces in space forms

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Abstract: Abstract Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.
PubDate: 20231201
DOI: 10.1007/s1336602200664w

 Logarithmic Ahypergeometric series $${\text {I}}\! {\text {I}}$$

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Abstract: Abstract In this paper, following (Saito in Int J Math 31(13):2050110, 2020), we continue to develop the perturbing method of constructing logarithmic series solutions to a regular Ahypergeometric system. Fixing a fake exponent of an Ahypergeometric system, we consider some spaces of linear partial differential operators with constant coefficients. Comparing these spaces, we construct a fundamental system of series solutions with the given exponent by the perturbing method. In addition, we give a sufficient condition for a given fake exponent to be an exponent. As important examples of the main results, we give fundamental systems of series solutions to AomotoGel’fand systems and to Lauricella’s \(F_C\) systems with special parameter vectors, respectively.
PubDate: 20231201
DOI: 10.1007/s13366022006695

 A note on a Cohentype theorem for Artinian modules

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Abstract: Abstract In this paper, we prove that a finitely embedded Rmodule M is Artinian if and only if for every prime ideal \(\mathfrak {p}\) of R with \((0:_RM)\subseteq \mathfrak {p}\) , there exists a submodule \(N^\mathfrak {p}\) of M such that \(M/N^\mathfrak {p}\) is finitely embedded and \(M[\mathfrak {p}]\subseteq N^\mathfrak {p}\subseteq (0:_M\mathfrak {p})\) , where \(M[\mathfrak {p}]=\bigcap \nolimits _{s\in R {\setminus } \mathfrak {p}}s(0:_M\mathfrak {p}).\)
PubDate: 20231201
DOI: 10.1007/s1336602200671x

 On the slope inequalities for extremal curves

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Abstract: Abstract The present paper concerns the question of the violation of the rth inequality for extremal curves in \(\mathbb {P}^r\) , posed in [KM]. We show that the answer is negative in many cases (Theorem 4.13 and Corollary 4.14). The result is obtained by a detailed analysis of the geometry of extremal curves and their canonical model. As a consequence, we show that particular curves on a Hirzebruch surface do not violate the slope inequalities in a certain range (Theorem 6.4).
PubDate: 20231201
DOI: 10.1007/s1336602200662y

 Groups with almost Frattini closed subgroups

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Abstract: Abstract A subgroup H of a group G is said to be Frattini closed in G if either \(H=G\) or H is the intersection of all maximal subgroups of G containing H. The structure of a soluble group in which every subgroup is Frattini closed is known. In this paper, the behavior of a (generalized) soluble group G in which every subgroup is Frattini closed in a subgroup of finite index of G is studied. Among other results, it is proved that if G is a (generalized) soluble group and there exists a positive integer k such that every subgroup of G is Frattini closed in a subgroup of index at most k in G, then G contains a normal subgroup of finite index in which all subgroups are Frattini closed.
PubDate: 20231201
DOI: 10.1007/s13366022006677

 PBW filtration and monomial bases for Demazure modules in types A and C

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Abstract: Abstract We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements depends not on the type A or C of the Lie algebra but on the rank only.
PubDate: 20231201
DOI: 10.1007/s13366022006600

 Some applications of signed distances in triangles and circles

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Abstract: Abstract We present generalizations of Carnot’s theorem and of the classical Erdös–Mordell inequality.
PubDate: 20231201
DOI: 10.1007/s13366022006588

 Lattice zonotopes of degree 2

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Abstract: Abstract The Ehrhart polynomial \({\text {ehr}}_P (n)\) of a lattice polytope P gives the number of integer lattice points in the nth dilate of P for all integers \(n\ge 0\) . The degree of P is defined as the degree of its \(h^*\) polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solidangles and the lattice width, we provide a characterization of all 3dimensional zonotopes of degree 2.
PubDate: 20231201
DOI: 10.1007/s13366022006659

 Quadrilateral reptiles

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Abstract: Abstract A polygon P is called a reptile, if it can be decomposed into \(k\ge 2\) nonoverlapping and congruent polygons similar to P. We prove that if a cyclic quadrilateral is a reptile, then it is a trapezoid. Comparing with results of Betke and Osburg we find that every convex reptile is a triangle or a trapezoid.
PubDate: 20231201
DOI: 10.1007/s1336602200663x
