Computing homomorphisms between holonomic D-modules

Let K be a subfield of the complex numbers, and let D be the Weyl algebra of K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic left D-modules we present an algorithm that computes explicit generators for the finite dimensional vector space hom_D(M,N). This enables us to answer algorithmically whether two given holonomic modules are isomorphic. More generally, our algorithm can be used to get explicit generators for ext^i_D(M,N) for any i.Comment: 30 pages, AMS-LaTex, uses verbatim,amsmath,latexsym,amssymb,amsbsy,diagram

A computational approach to the D-module of meromorphic functions

Let $D$ be a divisor in ${\bf C}^n$. We present methods to compare the ${\mathcal D}$-module of the meromorphic functions ${\mathcal O}[* D]$ to some natural approximations. We show how the analytic case can be treated with computations in the Weyl algebra.Comment: 11 page

Explicit calculations in rings of differential operators

We use the notion of a standard basis to study algebras of linear differential operators and finite type modules over these algebras. We consider the polynomial and the holomorphic cases as well as the formal case. Our aim is to demonstrate how to calculate classical invariants of germs of coherent (left) modules over the sheaf D of linear differential operators over Cn. The main invariants we deal with are: the characteristic variety, its dimension and the multiplicity of this variety at a point of the cotangent space. In the final chapter we shall study more refined invariants of D-modules linked to the question of irregularity: The slopes of a D-module along a smooth hypersurface of the base space.Dans ce cours on développe la notion de base standard, en vue d’étudier les algèbres d’opérateurs différentiels linéaires et les modules de type fini sur ces algèbres. On considère le cas des coefficients polynomiaux, des coefficients holomorphes ainsi que le cas des algèbres d’opérateurs à coefficients formels. Notre but est de montrer comment les bases standards permettent de calculer certains invariants classiques des germes de modules (à gauche) cohérents sur le faisceaux D des opérateurs différentiels linéaires sur Cn. Les principaux invariants que nous examinons sont : la variét´é caractéristique, sa dimension et sa multiplicité en un point du fibré cotangent. Dans le dernier chapitre nous étudions des invariants plus fins des D-modules qui sont reliés aux questions d’irrégularité : les pentes d’un D-module, le long d’une hypersurface lisse.Dirección General de Enseñanza Superior e Investigación CientíficaMinisterio de Ciencia y TecnologíaPlan Andaluz de Investigación (Junta de Andalucía

Blow-up algebras in Algebra, Geometry and Combinatorics

[eng] The primary topic of this thesis lies at the crossroads of Commutative Algebra and its interactions with Algebraic Geometry and Combinatorics. It is mainly focused around the following themes: I Defining equations of blow-up algebras. II Study of rational maps via blow-up algebras. III Asymptotic properties of the powers of edge ideals of graphs. We are primarily interested in questions that arise in geometrical or combinatorial contexts and try to understand how their possible answers manifest in various algebraic structures or invariants. There is a particular algebraic object, the Rees algebra (or blow-up algebra), that appears in many constructions of Commutative Algebra, Algebraic Geometry, Geometric Modeling, Computer Aided Geometric Design and Combinatorics. The workhorse and main topic of this doctoral dissertation has been the study of this algebra under various situations. The Rees algebra was introduced in the field of Commutative Algebra in the famous paper [45]. Since then, it has become a central and fundamental object with numerous applications. The study of this algebra has been so fruitful that it is difficult to single out particular results or papers, instead we refer the reader to the books [52] and [53] to wit the “landscape of blow-up algebras”. From a geometrical point of view, the Rees algebra corresponds with the bi-homogeneous coordinate ring of two fundamental objects: the blow-up of a projective variety along a subvariety and the graph of a rational map between projective varieties (see [27, §II.7]). Therefore, the importance of finding the defining equations of the Rees algebra is probably beyond argument. This is a problem of tall order that has occupied commutative algebraists and algebraic geometers, and despite an extensive effort (see [6, 8, 15–19, 30, 37, 39–43, 51]), it remains open even in the case of polynomial rings in two variables. In [10], Chapter 2 of this dissertation, we use the theory of D-modules to describe the defining ideal of the Rees algebra in the case of a parametrization of a plane curve. The study of rational and birational maps is classical in the literature from both an algebraic and geometric point of view, and it goes back to the work of Cremona [20], at least. A relatively new idea, probably first used in [31], is to look at the syzygies of the base ideal of a rational map to determine birationality. This algebraic method for studying rational maps has now become an active research topic (see [7, 22, 23, 28, 29, 38, 44, 46, 47]). In a joint work with Buse´ and D’Andrea [9], Chapter 3 of this dissertation, we introduce a new algebra that we call the saturated special fiber ring, which turns out to be an important tool to analyze the degree of a rational map. Later, in [11], Chapter 4 of this dissertation, we compute the multiplicity of this new algebra in the case of perfect ideals of height two, which, in particular, provides an effective method to determine the degree of a rational map having those ideals as base ideal. Often a good tactic to approach a challenging problem is to go all the way up to a generic case and then find sufficient conditions for the specialization to keep some of the main features of the former. The procedure depends on taking a dramatic number of variables to allow modifying the given data into a generic shape, and usually receives the name of specialization. This method is seemingly due to Kronecker and was quite successful in the hands of Hurwitz ([34]) in establishing a new elegant theory of elimination and resultants. More recent instances where specialization is used are, e.g., [32], [33], [50], [48]. In a joint work with Simis [14], Chapter 5 of this dissertation, we consider the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The Rees algebra of the edge ideal of a graph is a well studied object (see [24, 25, 49, 54–57]), that relates combinatorial properties of a graph with algebraic invariants of the powers of its edge ideal. For the Rees algebra of 1 2 YAIRON CID RUIZ the edge ideal of a bipartite graph, in [12], Chapter 6 of this dissertation, we compute the universal Gro¨ bner basis of its defining equations and its total Castelnuovo-Mumford regularity as a bigraded algebra. It is a celebrated result that the regularity of the powers of a homogeneous ideal is asymptotically a linear function (see [21, 36]). Considerable efforts have been put forth to understand the form of this asymptotic linear function in the case of edge ideals (see [1–5, 26, 35]). In a joint work with Jafari, Picone and Nemati [13], Chapter 7 of this dissertation, for bicyclic graphs, i.e. graphs containing exactly two cycles, we characterize the regularity of its edge ideal in terms of the induced matching number and determine the previous asymptotic linear function in special cases. The basic outline of this thesis is as follows. In Chapter 1, we recall some preliminary results and definitions to be used along this work. Then, the thesis is divided in three different parts. The first part corresponds with the theme “ I Defining equations of blow-up algebras” and consists of Chapter 2. The second part corresponds with the theme “ II Study of rational maps via blow-up algebras” and consists of Chapter 3, Chapter 4 and Chapter 5. The third part corresponds with the theme “ III Asymptotic properties of the powers of edge ideals of graphs” and consists of Chapter 6 and Chapter 7. The common thread and main tool in the three parts of this thesis is the use of blow-up algebras. References [1] A. Alilooee and A. Banerjee, Powers of edge ideals of regularity three bipartite graphs, J. Commut. Algebra 9 (2017), no. 4, 441–454. [2] A. Alilooee, S. Beyarslan, and S. Selvaraja, Regularity of powers of unicyclic graphs, Rocky Mountain J. Math. (2018). Advance publication. [3] A. Banerjee, The regularity of powers of edge ideals, J. Algebraic Combin. 41 (2015), no. 2, 303–321. [4] A. Banerjee, S. Beyarslan, and H. T. Ha, Regularity of edge ideals and their powers, arXiv preprint arXiv:1712.00887 (2017). [5] S. Beyarslan, H. T. Ha`, and T. N. Trung, Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095. [6] J. A. Boswell and V. Mukundan, Rees algebras and almost linearly presented ideals, J. Algebra 460 (2016), 102–127. [7] N. Botbol, L. Buse´, M. Chardin, S. H. Hassanzadeh, A. Simis, and Q. H. Tran, Effective criteria for bigraded birational maps, J. Symbolic Comput. 81 (2017), 69–87. [8] L. Buse´, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra 321 (2009), no. 8, 2317–2344. [9] L. Buse´, Y. Cid-Ruiz, and C. D’Andrea, Degree and birationality of multi-graded rational maps, ArXiv e-prints (May 2018), available at 1805.05180. [10] Y. Cid-Ruiz, A D-module approach on the equations of the Rees algebra, to appear in J. Commut. Algebra (2017). arXiv:1706.06215. [11] , Multiplicity of the saturated special fiber ring of height two perfect ideals, ArXiv e-prints (July 2018). 1807.03189. [12] , Regularity and Gro¨ bner bases of the Rees algebra of edge ideals of bipartite graphs, Le Matematiche 73 (2018), no. 2, 279–296. [13] Y. Cid-Ruiz, S. Jafari, N. Nemati, and B. Picone, Regularity of bicyclic graphs and their powers, to appear in J. Algebra Appl. (2018). arXiv:1802.07202. [14] Y. Cid-Ruiz and A. Simis, Degree of rational maps via specialization, arXiv preprint arXiv:1901.06599 (2019). [15] T. Cortadellas Ben´ıtez and C. D’Andrea, Rational plane curves parameterizable by conics, J. Algebra 373 (2013), 453–480. [16] , Minimal generators of the defining ideal of the Rees algebra associated with a rational plane parametrization with µ = 2, Canad. J. Math. 66 (2014), no. 6, 1225–1249. [17] , The Rees algebra of a monomial plane parametrization, J. Symbolic Comput. 70 (2015), 71–105. [18] D. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), no. 1-3, 23–36. [19] D. Cox, J. W. Hoffman, and H. Wang, Syzygies and the Rees algebra, J. Pure Appl. Algebra 212 (2008), no. 7, 1787–1796. [20] L. Cremona, Sulle trasformazioni geometriche delle figure piane, Mem. Acad. Bologna 2 (1863), no. 2, 621–630. [21] S. D. Cutkosky, J. Herzog, and N. V. Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), no. 3, 243–261. MR1711319 [22] A. V. Doria, S. H. Hassanzadeh, and A. Simis, A characteristic-free criterion of birationality, Adv. Math. 230 (2012), no. 1, 390–413. [23] D. Eisenbud and B. Ulrich, Row ideals and fibers of morphisms, Michigan Math. J. 57 (2008), 261–268. Special volume in honor of Melvin Hochster. [24] L. Fouli and K.-N. Lin, Rees algebras of square-free monomial ideals, J. Commut. Algebra 7 (2015), no. 1, 25–54. [25] I. Gitler, C. Valencia, and R. H. Villarreal, A note on the Rees algebra of a bipartite graph, J. Pure Appl. Algebra 201 (2005), no. 1-3, 17–24. [26] H. T. Ha`, Regularity of squarefree monomial ideals, Connections between algebra, combinatorics, and geometry, 2014, pp. 251–276. [27] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. [28] S. H. Hassanzadeh and A. Simis, Plane Cremona maps: saturation and regularity of the base ideal, J. Algebra 371 (2012), 620–652. [29] , Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs, J. Algebra 478 (2017), 220–236. [30] J. Hong, A. Simis, and W. V. Vasconcelos, On the homology of two-dimensional elimination, J. Symbolic Comput. 43 (2008), no. 4, 275–292. [31] K. Hulek, S. Katz, and F.-O. Schreyer, Cremona transformations and syzygies., Math. Z. 209 (1992), no. 3, 419–443. [32] C. Huneke and B. Ulrich, Residual intersections, J. Reine Angew. Math. 390 (1988), 1–20. [33] , Generic residual intersections, Commutative algebra (Salvador, 1988), 1990, pp. 47–60. [34] A. Hurwitz, U¨ ber die Tra¨ gheitsformen eines algebraischen Moduls., Annali di Mat. (3) 20 (1913), 113–151 (Italian). [35] A. V. Jayanthan, N. Narayanan, and S. Selvaraja, Regularity of powers of bipartite graphs, Journal of Algebraic Combinatorics (2017May). BRIEF RESUME OF THE PHD THESIS 3 [36] V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. 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An introduction to constructive algebraic analysis and its applications

This text is an extension of lectures notes I prepared for les Journées Nationales de Calcul Formel held at the CIRM, Luminy (France) on May 3-7, 2010. The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly to algebraic D-modules, its applications to mathematical systems theory and its implementations in computer algebra systems such as Maple or GAP4. Since algebraic analysis is a mathematical theory which uses different techniques coming from module theory, homological algebra, sheaf theory, algebraic geometry, and microlocal analysis, it can be difficult to enter this fascinating new field of mathematics. Indeed, there are very few introducing texts. We are quickly led to Björk's books which, at first glance, may look difficult for the members of the symbolic computation community and for applied mathematicians. I believe that the main issue is less the technical difficulty of the existing references than the lack of friendly introduction to the topic, which could have offered a general idea of it, shown which kind of results and applications we can expect and how to handle the different computations on explicit examples. To a very small extent, these lectures notes were planned to fill this gap, at least for the basic ideas of algebraic analysis. Since, we can only teach well what we have clearly understood, I have chosen to focus on my work on the constructive aspects of algebraic analysis and its applications.Ce texte est une extension des notes de cours que j'ai préparés pour les les Journées Nationales de Calcul Formel qui ont eu lieu au CIRM, Luminy (France) du 3 au 7 Mai 2010. Le but principal de ce cours était d'introduire la communauté française du calcul formel à l'analyse algébrique constructive, et particulièrement à la théorie des D-modules algébriques, à ses applications à la théorie mathématique des systèmes et à ses implantations dans des logiciels de calcul formel tels que Maple ou GAP4. Parce que l'analyse algébrique est une théorie mathématique qui utilise différentes techniques venant de la théorie des modules, de l'algèbre homologique, de la théorie des faisceaux, de la géométrie algébrique et de l'analyse microlocale, il peut être difficile d'entrer dans ce domaine, nouveau et fascinant, des mathématiques. En effet, il existe peu de textes introductifs. Nous sommes rapidement conduits aux livres de Björk qui, à première vue, peuvent sembler difficiles aux membres de la communauté de calcul formel et aux mathématiciens appliqués. Je pense que le problème vient moins de la difficulté technique de la littérature existante que du manque d'introductions pédagogiques qui donnent une idée globale du domaine, montrent quels types de résultats et d'applications on peut attendre et qui développent les différents calculs à mener sur des exemples explicites. A leur humble niveau, ces notes de cours ont pour but de combler ce manque, tout du moins en ce qui concerne les idées de base de l'analyse algébrique. Puisque l'on ne peut enseigner bien que les choses que l'on a bien comprises, j'ai choisi de restreindre cette introduction à mes travaux sur les aspects constructifs de l'analyse algébrique et sur ses applications