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A. V. Jayanthan

Bio: A. V. Jayanthan is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Ideal (ring theory) & Local ring. The author has an hindex of 14, co-authored 54 publications receiving 530 citations. Previous affiliations of A. V. Jayanthan include Tata Institute of Fundamental Research & Indian Institutes of Technology.


Papers
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Journal ArticleDOI
TL;DR: For bipartite graphs, this paper obtained upper bounds on the polarization of the ideal edge ideal for all vertices of a simple graph G and the corresponding edge ideal ideal ideal I(G).
Abstract: Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all $$s \ge 1$$ , we obtain upper bounds for $${\text {reg}}(I(G)^s)$$ for bipartite graphs. We then compare the properties of G and $$G'$$ , where $$G'$$ is the graph associated with the polarization of the ideal $$(I(G)^{s+1} : e_1\cdots e_s)$$ , where $$e_1,\cdots , e_s$$ are edges of G. Using these results, we explicitly compute $${\text {reg}}(I(G)^s)$$ for several subclasses of bipartite graphs.

70 citations

Posted Content
TL;DR: For bipartite graphs, the authors obtained upper bounds for reg$(I(G)^s)$ for the polarization of the ideal of a simple simple graph and the corresponding edge ideal.
Abstract: Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph associated with the polarization of the ideal $(I(G)^{s+1} : e_1\cdots e_s)$, where $e_1,\ldots e_s$ are edges of $G$. Using these results, we explicitly compute reg$(I(G)^s)$ for several subclasses of bipartite graphs.

43 citations

Journal ArticleDOI
TL;DR: In this paper, the Hilbert coefficients for the fiber cone F ( I ) of an m -primary ideal I in a Cohen-Macaulay local ring ( R, m ) were given in terms of certain Hilbert coefficients.

42 citations

Journal ArticleDOI
01 Jun 2019
TL;DR: In this article, it was shown that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularities of individual graphs.
Abstract: We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid (Electron J Comb 20(4), 2013). We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of block-graphs. As a consequence, we obtain sharp upper and lower bounds for a class of trees called lobsters.

36 citations

Journal ArticleDOI
TL;DR: In this article, the Castelnuovo-Mumford regularity of S/JG when JG is a binomial edge ideal is computed for simple graphs on n vertices.
Abstract: Let G be a finite simple graph on n vertices and JG denote the corresponding binomial edge ideal in S=K[x1,…,xn,y1,…,yn]. We compute the Castelnuovo-Mumford regularity of S/JG when JG is th...

32 citations


Cited by
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Book
01 Jan 2006
TL;DR: In this paper, the authors define the integral closure of rings and define a table of basic properties including separation, separationability, separation of rings, and normal homomorphisms, and the Briancon-Skoda theorem.
Abstract: Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

826 citations

Book
25 Aug 2010
TL;DR: In this paper, the theory of Hilbert functions of modules over local rings is discussed, and the authors present a unified approach to give self-contained and easier proofs for the theory.
Abstract: In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.

119 citations

Journal ArticleDOI
TL;DR: In this paper, the Hilbert coefficients for the fiber cone F ( I ) of an m -primary ideal I in a Cohen-Macaulay local ring ( R, m ) were given in terms of certain Hilbert coefficients.

42 citations

Journal ArticleDOI
TL;DR: In this article, the apolar ideal of a polynomial of degree δ has a minimal generator of degree d if and only if it is a limit of direct sums.
Abstract: A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct sum, in terms of the apolar ideal of the polynomial. We prove that the apolar ideal of a polynomial of degree $d$ strictly depending on all variables has a minimal generator of degree $d$ if and only if it is a limit of direct sums.

41 citations