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C P Anil Kumar

Publications and source records attributed to C P Anil Kumar.

13 recordsLinked to original sources
arXiv2021

On Very Generic Discriminantal Arrangements

In this article we prove two main results. Firstly, we show that any six-line arrangement, consisting of three pairs of mutually perpendicular lines, does not give rise to a "very generic or sufficiently general" discriminantal arrangement in the sense of C. A. Athanasiadis \cite{MR1720104}. We give two proofs of the first result. The second result is as follows. The codimension-one boundary faces of (a region) a convex cone of a very generic discriminantal arrangement has not been characterized and is not known even though the intersection lattice of a very generic discriminantal arrangement is known. So secondly, we show that the number of simplex cells of the very generic hyperplane arrangement $\mathcal{H}^m_n=\{H_i:\underset{j=1}{\overset{m}{\sum}}a_{ij}x_j=c_i,1\leq i\leq n\}$ may not be not precisely equal to the number of codimension-one boundary hyperplanes of $\mathbb{R}^n$ of the convex cone $C$ containing $(c_1,c_2,\ldots,c_n)$ in the associated very generic discriminantal arrangement. That is, for $1\leq i_1<i_2<\ldots<i_m<i_{m+1}\leq n$, if $Δ^m H_{i_1}H_{i_2}\ldots H_{i_m}H_{i_{m+1}}$ is a simplex cell of the hyperplane arrangement $\mathcal{H}^m_n$ then it need not give rise to a codimension-one boundary hyperplane of the convex cone $C$ containing $(c_1,c_2,\ldots,c_n)$ in the associated very generic discriminantal arrangement. We finally mention an interesting open-ended remark before the appendix section. In the appendix section we give a self contained exposition and describe combinatorially the intersection lattice of a (Zariski open and dense) class of "very generic or sufficiently general" discriminantal arrangements. As a consequence, we give a geometric description of the lattice elements as sets of concurrencies of the hyperplane arrangements which give the same "very generic or sufficiently general" discriminantal arrangement.

math.CO↗
arXiv2021

On the Positivity Conjecture for Finite Abelian p-Groups

For a partition $\underlineλ = (λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^{ρ_3}>\ldots>λ_{k}^{ρ_k})$ and its associated finite $\mathcal{R}$-module $\mathcal{A}_{\underlineλ}=\underset{i=1}{\overset{k}{\oplus}} (\mathcal{R}/π^{λ_i}\mathcal{R})^{ρ_i}$, where $\mathcal{R}$ is a discrete valuation ring, with maximal ideal generated by a uniformizing element $π$, having finite residue field ${\bf k}=\mathcal{R}/π\mathcal{R}\cong \mathbb{F}_q$, the number of orbits of pairs $n_{\underlineλ}(q)= \mid \mathcal{G}_{\underlineλ}\backslash \big(\mathcal{A}_{\underlineλ}\times \mathcal{A}_{\underlineλ}\big)\mid$ for the diagonal action of the automorphism group $\mathcal{G}_{\underlineλ}= Aut(\mathcal{A}_{\underlineλ})$, is a polynomial in $q$ with integer coefficients. Positivity conjecture states that these coefficients are in fact non-negative. In this article, we prove this conjecture.

math.CO↗
arXiv2021

On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point

For a positive integer $n\geq 3$, the sides and diagonals of a convex $n$-gon divide the interior of the convex $n$-gon into finitely (polynomial in $n$) many regions bounded by them. In this article, we associate to every region a unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as $2$-standard consecutive cycle) by studying point arrangements in the plane. Then we find that there are more (exponential in $n$) number of such cycles leading to the conclusion that not every region labelled by a cycle appears in every convex $n$-gon. In fact most of them do not occur in any given single convex $n$-gon. Later in the main theorem of this article we characterize combinatorially those cycles (defined as definite cycles) whose corresponding regions occur in every convex $n$-gon and those cycles (defined as indefinite cycles) whose corresponding regions do not occur in every convex $n$-gon. As a consequence we characterize those one point extensions of a uniform rank $3$ convex oriented matroid for which the one point extension is reducible by the, one point, when it lies inside the convex hull.

math.CO↗
arXiv2021

On the Enumeration of a Certain Type of Hyperplane Arrangements

In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discriminantal arrangement. Later we observe that the restriction we impose on the type of hyperplane arrangements is a mild one and that this conditional restriction is quite generic. Moreover the restriction is defined in terms of a normal system being concurrency free which is a generic condition. We also discuss two examples of normal systems which are not concurrency free in the last section and enumerate the number of isomorphism classes.

math.CO↗
arXiv2021

On the Generic Point Arrangements in Euclidean Space and Stratification of the Totally Nonzero Grassmannian

In this article, for positive integers $n\geq m\geq 1$, the parameter spaces for the isomorphism classes of the generic point arrangements of cardinality $n$, and the antipodal point arrangements of cardinality $2n$ in the Eulidean space $\mathbb{R}^m$ are described using the space of totally nonzero Grassmannian $Gr^{tnz}_{mn}(\mathbb{R})$. A stratification $\mathcal{S}^{tnz}_{mn}(\mathbb{R})$ of the totally nonzero Grassmannian $Gr^{tnz}_{mn}(\mathbb{R})$ is mentioned and the parameter spaces are respectively expressed as quotients of the space $\mathcal{S}^{tnz}_{mn}(\mathbb{R})$ of strata under suitable actions of the symmetric group $S_n$ and the semidirect product group $(\mathbb{R}^*)^n\rtimes S_n$. The cardinalities of the space $\mathcal{S}^{tnz}_{mn}(\mathbb{R})$ of strata and of the parameter spaces $S_n\backslash \mathcal{S}^{tnz}_{mn}(\mathbb{R}), ((\mathbb{R}^*)^n\rtimes S_n)\backslash \mathcal{S}^{tnz}_{mn}(\mathbb{R})$ are enumerated in dimension $m=2$. Interestingly enough, the enumerated value of the isomorphism classes of the generic point arrangements in the Euclidean plane is expressed in terms of the number theoretic Euler-totient function. The analogous enumeration questions are still open in higher dimensions for $m\geq 3$.

math.GM↗
arXiv2020

On the Triangles in Certain Types of Line Arrangements

In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been described for both the types and some properties of the nomenclature have been proved. Later using the nomenclature we describe the triangles present in both types of line arrangements in Theorems $A,B$. We also prove that the set of triangles uniquely determine, in a certain precise sense, the line arrangements with global cyclicity and not the infinity type line arrangements where counter examples have been provided. In Theorem $9.1$, given a nomenclature, we characterize when a particular line symbol in the nomenclature is a line at infinity for the arrangement determined by the nomenclature.

math.GM↗
arXiv2020

A Combinatorial Identity for the p-Binomial Coefficient Based on Abelian Groups

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $\mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underlineλ$, which is a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian $p$-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in $p$ with non-negative integer coefficients.

math.CO↗
arXiv2020

On Rational Sets in Euclidean Spaces and Spheres

IFor a positive rational $l$, we define the concept of an $l$-elliptic and an $l$-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite $l$-hyperbolic and $l$-ellitpic rationals subsets of the real line and unit circle. For the case of a circle, we prove that the existence of such sets depends on the positivity of ranks of certain associated elliptic curves. We also determine the closures of such sets which are maximal in case they are not dense. In higher dimensions, we show the existence of $l$-ellitpic and $l$-hyperbolic rational infinite sets in unit spheres and Euclidean spaces for certain values of $l$ which satisfy a weaker condition regarding the existence of elements of order more than two, than the positivity of the ranks of the same associated elliptic curves. We also determine their closures. A subset $T$ of the $k$-dimensional unit sphere $S^k$ has an antipodal pair if both $x,-x\in T$ for some $x\in S^k$. In this article, we prove that there does not exist a dense rational set $T\subset S^2$ which has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of general type. We actually show that the existence of such a dense rational set in $S^k$ is equivalent to the existence of a dense $2$-hyperbolic rational set in $S^k$ which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space $\mathbb{R}^k$.

math.NT↗
arXiv2020

On the Surjectivity of Certain Maps III: The Unital Set Condition

In this article, for generalized projective spaces with any weights, we prove four main theorems in three different contexts where the Unital Set Condition USC (Definition $2.8$) on ideals is further examined. In the first context we prove, in the first main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal $\mathcal{I}=\underset{i=1}{\overset{k}{\prod}}\mathcal{I}_k$ with a given factorization into mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq k$ where $\mathcal{I}$ satisfies the USC, using the key concept of choice multiplier hypothesis (Definition $4.10$) which is satisfied. In the second context, for a positive $k$, we prove in the second main Theorem $Λ$, the surjectivity of the reduction map $SP_{2k}(\mathcal{R})\rightarrow SP_{2k}(\frac{\mathcal{R}}{\mathcal{I}})$ of strong approximation type for a ring $\mathcal{R}$ quotiented by an ideal $\mathcal{I}$ which satisfies the USC. In the third context, for a positive integer $k$, we prove in the thrid main Theorem $Ω$, the surjectivity of the map from special linear group of degree $(k+1)$ to the product of generalized projective spaces of $(k+1)$-mutually co-maximal ideals $\mathcal{I}_j,0\leq j\leq k$ associating the $(k+1)$-rows or $(k+1)$-columns, where the ideal $\mathcal{I}=\underset{j=0}{\overset{k}{\prod}}\mathcal{I}_j$ satisfies the USC. In the fourth main Theorem $Σ$, for a positive integer $k$, we prove the surjectivity of the map from the symplectic group of degree $2k$ to the product of generalized projective spaces of $(2k)$-mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq 2k$ associating the $(2k)$-rows or $(2k)$-columns where the ideal $\mathcal{I}=\underset{j=1}{\overset{2k}{\prod}}\mathcal{I}_j$ satisfies the USC. The answers to Questions [1.1,1.2,1.3] in a greater generality are not known.

math.NT↗
arXiv2020

On a Conjecture of Kelly on $(1,3)$-representation of Sylvester Gallai Designs

We give an exact criterion of a conjecture of L.M.Kelly to hold true which is stated as follows. If there is a finite family $Σ$ of mutually skew lines in $\mathbb{R}^l,l\geq 4$ such that the three dimensional affine span (hull) of every two lines in $Σ$, contains at least one more line of $Σ$, then we have that $Σ$ is entirely contained in a three dimensional space if and only if the arrangement of affine hulls is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, that is, for $(2,5)$-representations of sylvester-gallai designs in $\mathbb{R}^6$, which is answered in the last section.

math.CO↗
arXiv2019

On a Projective Space Invariant of a Co-torsion Module of Rank Two over a Dedekind Domain

For a Dedekind domain $\mathcal{O}$ and a rank two co-torsion module $M\subseteq \mathcal{O}^2$ with invariant factor ideals $\mathcal{L}\supseteq \mathcal{K}$ in $\mathcal{O}$, that is, $\frac{\mathcal{O}^2}{M}\cong \frac{\mathcal{O}}{\mathcal{L}}\oplus \frac{\mathcal{O}}{\mathcal{K}}$ we associate a new projective space invariant element in $\mathbb{PF}^1_{\mathcal{I}}$ where $\mathcal{I}$ is given by the ideal factorization $\mathcal{K} = \mathcal{L}\mathcal{I}$ in $\mathcal{O}$. This invariant element along with the invariant factor ideals determine the module $M$ completely as a subset of $\mathcal{O}^2$. As a consequence, projective spaces associated to ideals in $\mathcal{O}$ can be used to enumerate such modules. We compute the zeta function associated to such modules in terms of the zeta function of the one dimensional projective spaces for the ring $\mathcal{O}_K$ of integers in a number field $K/\mathbb{Q}$ and relate them to Dedekind zeta function. Using the projective spaces as parameter spaces, we re-interpret the Chinese remainder reduction isomorphism $\mathbb{PF}^1_{\mathcal{I}} \rightarrow \underset{i=1}{\overset{l}{\prod}} \mathbb{PF}^1_{\mathcal{I}_i}$ associated to a factorization of an ideal $\mathcal{I}=\underset{i=1}{\overset{l}{\prod}} \mathcal{I}_i$ into mutually co-maximal ideals $\mathcal{I}_i,1\leq i\leq l$ in terms of the intersection of associated modules arising from the projective space elements.

math.NT↗
arXiv2019

On the Endomorphism Semigroups of Extra-special $p$-groups and Automorphism Orbits

For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.

math.GR↗