Algebraic Matroids and Frobenius Flocks

Total Page:16

File Type:pdf, Size:1020Kb

Algebraic Matroids and Frobenius Flocks Algebraic matroids and Frobenius flocks Citation for published version (APA): Bollen, G. P., Draisma, J., & Pendavingh, R. A. (2017). Algebraic matroids and Frobenius flocks. arXiv, (1701.06384v2), 1-21. [1701.06384v2]. http://arxiv.org/abs/1701.06384v2 Document status and date: Published: 23/01/2017 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 01. Oct. 2021 ALGEBRAIC MATROIDS AND FROBENIUS FLOCKS GUUS P. BOLLEN, JAN DRAISMA, AND RUDI PENDAVINGH Abstract. We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindstr¨omvaluation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindstr¨omvaluation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations. 1. Introduction Several years before Whitney defined matroids in [22], Van der Waerden [21] recognized the commonality between algebraic and linear independence which is the defining property of matroids (see [17, Section 39.10b]). Today, linear matroids are the subject of some of the deepest theorems in combinatorics. By contrast, our understanding of the algebraic matroid is much less advanced. Algebraic matroids seem forebodingly hard to deal with, compared to linear matroids. Deciding if a matroid has a linear representation over a given field amounts to solving a system of polynomial equations over that field, which makes the linear representation problem decidable over algebraically closed fields (for the situation of representability over Q, see [20]). We do not know of any procedure to decide if a matroid has an algebraic representation; for example, it is an open problem if the Tic-Tac-Toe matroid, a matroid of rank 4 on 9 elements, is algebraic [7]. It is possible to describe the set of all linear representations of a given matroid as being derived from a `universal' matroid representation of that matroid over a partial field [15]. We tried to find a full set of invariants to classify (equivalence classes of) algebraic representations of the tiny uniform matroid U2;3, and failed. Since algebraic representations appear so difficult to deal with in full detail, we are pleased to report that we found a nontrivial invariant of algebraic matroid representations in positive characteristic, taking inspiration in particular from the work of Lindstr¨om[12]. Throughout the paper, we use the language of algebraic geometry to describe matroid repre- sentations, and we work over an algebraically closed field K. An irreducible algebraic variety X ⊆ KE determines a matroid M(X) with ground set E by declaring a set I ⊆ E independent I if the projection of X on K is dominant, that is, if the closure of fxI : x 2 Xg in the Zariski topology equals KI . In the special case that X is a linear space, M(X) is exactly the matroid represented by the columns of any matrix whose rows span X. We next translate results of Ingleton and Lindstr¨omto the language of algebraic geometry. arXiv:1701.06384v2 [math.CO] 6 Feb 2017 Ingleton argued that if K has characteristic 0, then for any sufficiently general point x 2 X, the tangent space TxX of X at x will have the same dominant projections as X itself, so that then M(X) = M(TxX). Since such a sufficiently general point always exists | it suffices that x avoids finitely many hypersurfaces in X | a matroid which has an algebraic representation over K also has a linear representation over K, in the form of a tangent space TxX. Ingletons argument does not generalize to fields K of positive characteristic p. Consider the p 2 p variety X = V (x1 − x2) = fx 2 K : x1 − x2 = 0g, which represents the matroid on E = f1; 2g with independent sets ;; f1g; f2g. The tangent space of X at any x 2 X is TxX = V (x1), which represents a matroid in which f1g is a dependent set. Thus, M(X) =6 M(TxX) for all x 2 X. Lindstr¨omdemonstrated that for some varieties X this obstacle may be overcome by applying the Frobenius map F : x 7! xp to some of the coordinates, to derive varieties from X which GPB is funded by a grant from Stichting Computer Algebra Nederland. JD is partially funded by Vidi and Vici grants from the Netherlands Organisation for Scientific Research (NWO). 1 p represent the same matroid. In case of the counterexample X = V (x1 − x2) above, we could define X0 = f(x1;F (x2)) : x 2 Xg: Then M(X0) = M(X), and X0 = V (x1 − x2), so that E M(X0) = M(Tx0 X0) for some (in fact, all) x0 2 X0. In general for an X ⊆ K , if we fix a vector α 2 ZE, put αi αx := (F − xi)i E and αX := fαx : x 2 Xg; 2 then it can be argued that M(αX) = M(X). This gives additional options for finding a suitable tangent space. Lindstr¨omshowed in [12] that if X is any algebraic representation of the Fano matroid, then there necessarily exists an α so that M(X) = M(TξαX) for a sufficiently general ξ 2 αX. Thus any algebraic representation of the Fano matroid spawns a linear representation in the same characteristic. The choice of the matroid in Lindstr¨om's argument is not arbitrary. If we consider an algebraic representation X of the non-Fano matroid in characteristic 2, then there cannot be an α 2 ZE so that in a sufficiently general point ξ 2 αX we have M(X) = M(TξαX): then the tangent space would be a linear representation of the non-Fano matroid in characteristic 2, which is not possible. Thus any attempt to balance out the Frobenius actions α 2 ZE will be fruitless in this case. In the present paper, we consider the overall structure of the map α 7! M(Tξα αX), where for each α, ξα is the generic point of αX. A central result of this paper is that a sufficiently general E x 2 X satisfies M(TαxαX) = M(Tξα αX) for all α 2 Z (Theorem 38). This is nontrivial, as now x must a priori avoid countably many hypersurfaces, which one might think could cover all of X. Fixing such a general x, the assignment V : α 7! Vα := TαxαX is what we have named a E Frobenius flock: each Vα is a linear subspace of K of fixed dimension dim Vα = dim X =: d, and moreover E (FF1) Vα=i = Vα+ei ni for all α 2 Z and i 2 E; and E (FF2) Vα = 1Vα+1 for all α 2 Z . Here we wrote W=i := fwE i : w 2 W; wi = 0g and W ni := fwE i : w 2 W g; 1 is the all-one vector. − − In this paper, we are primarily concerned with properties of the matroids represented by the E vector spaces Vα | in particular, we want to understand when there exists an α 2 Z so that M(X) = M(Vα). We will call an assignment M : α 7! Mα of a matroid of rank d on E to each α 2 ZE a matroid flock if it satisfies E (MF1) Mα=i = Mα+ei ni for all α 2 Z and i 2 E; and E (MF2) Mα = Mα+1 for all α 2 Z . Clearly, any Frobenius flock V gives rise to a matroid flock by taking Mα = M(Vα). In particular, E each algebraic matroid representation X ⊆ K gives rise to a matroid flock M : α 7! M(TαxαX). This matroid flock does not depend on the choice of the very general point x 2 X, since M(TαxαX) = M(Tαx0 αX) for any two such very general points.
Recommended publications
  • The Hermite–Lindemann–Weierstraß Transcendence Theorem
    The Hermite–Lindemann–Weierstraß Transcendence Theorem Manuel Eberl March 12, 2021 Abstract This article provides a formalisation of the Hermite–Lindemann– Weierstraß Theorem (also known as simply Hermite–Lindemann or Lindemann–Weierstraß). This theorem is one of the crowning achieve- ments of 19th century number theory. The theorem states that if α1; : : : ; αn 2 C are algebraic numbers that are linearly independent over Z, then eα1 ; : : : ; eαn are algebraically independent over Q. Like the previous formalisation in Coq by Bernard [2], I proceeded by formalising Baker’s alternative formulation of the theorem [1] and then deriving the original one from that. Baker’s version states that for any algebraic numbers β1; : : : ; βn 2 C and distinct algebraic numbers αi; : : : ; αn 2 C, we have: α1 αn β1e + ::: + βne = 0 iff 8i: βi = 0 This has a number of immediate corollaries, e.g.: • e and π are transcendental • ez, sin z, tan z, etc. are transcendental for algebraic z 2 C n f0g • ln z is transcendental for algebraic z 2 C n f0; 1g 1 Contents 1 Divisibility of algebraic integers 3 2 Auxiliary facts about univariate polynomials 6 3 The minimal polynomial of an algebraic number 10 4 The lexicographic ordering on complex numbers 12 5 Additional facts about multivariate polynomials 13 5.1 Miscellaneous ........................... 13 5.2 Converting a univariate polynomial into a multivariate one . 14 6 More facts about algebraic numbers 15 6.1 Miscellaneous ........................... 15 6.2 Turning an algebraic number into an algebraic integer .... 18 6.3 Multiplying an algebraic number with a suitable integer turns it into an algebraic integer.
    [Show full text]
  • Commutative Algebra
    Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions .
    [Show full text]
  • Algebraic Matroids in Applications by Zvi H Rosen a Dissertation
    Algebraic Matroids in Applications by Zvi H Rosen A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Bernd Sturmfels, Chair Professor Lior Pachter Professor Yun S. Song Spring 2015 Algebraic Matroids in Applications Copyright 2015 by Zvi H Rosen 1 Abstract Algebraic Matroids in Applications by Zvi H Rosen Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bernd Sturmfels, Chair Algebraic matroids are combinatorial objects defined by the set of coordinates of an algebraic variety. These objects are of interest whenever coordinates hold significance: for instance, when the variety describes solution sets for a real world problem, or is defined using some combinatorial rule. In this thesis, we discuss algebraic matroids, and explore tools for their computation. We then delve into two applications that involve algebraic matroids: probability matrices and tensors from statistics, and chemical reaction networks from biology. i In memory of my grandparents יצחק אהרN וחנה רייזל דייוויס ז"ל! Isaac and Ann Davis יצחק יוסP ושרה רוזN ז"ל! Isaac and Sala Rosen whose courage and perseverance through adversity will inspire their families for generations. ii Contents Contents ii List of Figures iv List of Tables v 1 Introduction 1 1.1 Summary of Main Results . 1 1.2 Examples . 2 1.3 Definitions, Axioms, and Notation . 5 2 Computation 10 2.1 Symbolic Algorithm . 11 2.2 Linear Algebra . 11 2.3 Sample Computations for Applications . 15 3 Statistics: Joint Probability Matroid 25 3.1 Completability of Partial Probability Matrix .
    [Show full text]
  • 210 the Final Chapter, on Algebraic Independence of Transcendental
    210 BOOK REVIEWS [July The final chapter, on algebraic independence of transcendental numbers, contains an exposition of Siegel's method, by means of which it can be shown, under suitable circumstances, that alge­ braically independent solutions of a system of linear differential equa­ tions have algebraically independent values for algebraic argument. The general framework of the method is set out very clearly, and applications are made to (a) Lindemann's theorem, that if cei, • • • , an are algebraic numbers, linearly independent over the rationals, then e<*it . 9 ectn are algebraically independent over the rationals; and (b) Siegel's theorem, that if XQT^O is algebraic, the values JQ(XQ) and Jo(x0) of the Bessel function are algebraically independent over the rationals. The book ends with an interesting list of open problems which the author considers both important and attainable. W. J. LEVEQUE Applied analysis. By Cornelius Lanczos. Englewood Cliffs, Prentice Hall, 1956. 20+539 pp. $9.00. This is a systematic development of some of the ideas of Professor Lanczos in the field of applied analysis, rather than another textbook. Instead of the adjective "applied," the author suggests that "parexic"1 would be more appropriate. Parexic analysis lies between classical analysis and numerical analysis: it is roughly the theory of approxi­ mation by finite (or truncated infinite) algorithms. There is, therefore, some overlap with what is being called elsewhere "constructive theory of functions." However, this book is a very personal one, and has been influenced by the varied experiences of the author: in regular aca­ demic work, in government service, in industry, and now at the Dublin Institute for Advanced Study.
    [Show full text]
  • Algebraic Independence
    Algebraic Independence UGP : CS498A Report Advisor : Dr. Nitin Saxena Tushant Mittal Indian Institute of Technology, Kanpur Contents 1 Introduction 2 1.1 The Problem........................................2 1.2 Motivation.........................................2 1.3 Preliminary Definitions..................................3 2 Previous Work 4 2.1 Computability.......................................4 2.1.1 The \Brute force" Algorithm...........................4 2.2 Characteristic 0 (or large) fields.............................5 2.3 Witt-Jacobian Criterion..................................5 2.4 Generalizing the Jacobian.................................5 3 Dimension Reduction6 3.1 Notation..........................................6 3.2 The first approach....................................6 3.3 The k-gap.........................................7 3.3.1 Bivariate Case...................................7 4 New Criterion 9 4.1 Ideal Shrink........................................9 4.2 Criterion.......................................... 10 5 Conclusion and Future Directions 12 6 Acknowledgements 13 1 Chapter 1 Introduction 1.1 The Problem The concept of algebraic independence is a natural generalization of the familiar notion of linear dependence. More formally, Definition 1.1. A subset S of a field L is algebraically dependent over a subfield K if the elements of S satisfy a non-trivial polynomial equation with coefficients in K. ♦ A few concrete examples are : • Algebraic/Transcendental Numbers : L = C ;K = Q;S = fαg • Polynomials : L = F(x1; ··· ; xn);K = F;S = ff1; ··· ; fng The problem of testing algebraic independence is then, Given a set of polynomials ff1; ··· ; fng determine if they are algebraically dependent i.e does there 9 A 2 F[y1; ··· ; yn] such that A(f1; ··· ; fn) = 0. ( A is called its annihilating polynomial ). Examples 1. The set f = fx1; x2; ··· ; xkg is always algebraically independent. p p 2. Algebraic dependence depends on the underlying field, fx1 +x2; x1 +x2g is independent over p Q but is dependent over Fp with y2 − y1 as the annihilating polynomial.
    [Show full text]
  • Criteria of Algebraic Independence with Multiplicities and Interpolation Determinants
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 5, Pages 1845{1870 S 0002-9947(99)02216-3 Article electronically published on January 19, 1999 CRITERIA OF ALGEBRAIC INDEPENDENCE WITH MULTIPLICITIES AND INTERPOLATION DETERMINANTS MICHEL LAURENT AND DAMIEN ROY Abstract. We generalize Gel'fond's criterion of algebraic independence by taking into account the values of the derivatives of the polynomials, and show how the new criterion applies to proving results of algebraic independence using interpolation determinants. We also establish a new result of approximation of a transcendental number by algebraic numbers of bounded degree and size. It contains an earlier result of E. Wirsing and also a result announced by A. Durand. 1. Introduction The problem of adapting the method of Interpolation Determinants to questions of algebraic independence has been open since 1989 (see [4]). Very recently, a solution has been proposed in [7] and [8]. It combines techniques introduced in [5] in the context of linear forms in logarithms, together with approximation results of complex numbers by algebraic numbers with large degree. In this paper, we propose another solution, in some sense more traditional, because it relies on a criterion of algebraic independence which is nevertheless of a new type. We shall restrict ourselves to problems of algebraic independence in transcendence degree 1, for which the standard tool is a criterion of Gel’fond (see for example [1]). The principle of the method is very simple. Consider the interpolation deter- minant attached to a given situation of algebraic independence. Suppose that its entries belong to a field of transcendence degree 1 over Q, say for simplicity a field of rational functions Q(θ).
    [Show full text]
  • Algebraic Matroids and Frobenius Flocks
    ALGEBRAIC MATROIDS AND FROBENIUS FLOCKS GUUS P. BOLLEN, JAN DRAISMA, AND RUDI PENDAVINGH Abstract. We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindstr¨omvaluation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindstr¨omvaluation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations. 1. Introduction Several years before Whitney defined matroids in [23], Van der Waerden [22] recognized the commonality between algebraic and linear independence which is the defining property of matroids (see [18, Section 39.10b]). Today, linear matroids are the subject of some of the deepest theorems in combinatorics. By contrast, our understanding of algebraic matroids is much less advanced. Algebraic matroids seem forebodingly hard to deal with, compared to linear matroids. Deciding if a matroid has a linear representation over a given field amounts to solving a system of polynomial equations over that field, which makes the linear representation problem decidable over algebraically closed fields (for the situation of representability over Q, see [21]). We do not know of any procedure to decide if a matroid has an algebraic representation; for example, it is an open problem if the Tic-Tac-Toe matroid, a matroid of rank 5 on 9 elements, is algebraic [8].
    [Show full text]
  • Math 249A Fall 2010: Transcendental Number Theory
    Math 249A Fall 2010: Transcendental Number Theory A course by Kannan Soundararajan LATEXed by Ian Petrow September 19, 2011 Contents 1 Introduction; Transcendence of e and π α is algebraic if there exists p 2 Z[x], p 6= 0 with p(α) = 0, otherwise α is called transcendental . Cantor: Algebraic numbers are countable, so transcendental numbers exist, and are a measure 1 set in [0; 1], but it is hard to prove transcendence for any particular number. p p 2 Examples of (proported) transcendental numbers: e, π, γ, eπ, 2 , ζ(3), ζ(5) ::: p p 2 Know: e, π, eπ, 2 are transcendental. We don't even know if γ and ζ(5), ζ(7);::: are irrational or rational, and we know that ζ(3) is irrational, but not whether or not it is transcendental! Lioville showed that the number 1 X 10−n! n=1 is transcendental, and this was one of the first numbers proven to be transcen- dental. Theorem 1 (Lioville). If 0 6= p 2 Z[x] is of degree n, and α is a root of p, α 62 , then Q a C(α) α − ≥ : q qn Proof. Assume without loss of generality that α < a=q, and that p is irreducible. Then by the mean value theorem, p(α) − p(a=q) = (α − a=q)p0(ξ) 1 for some point ξ 2 (α; a=q). But p(α) = 0 of course, and p(a=q) is a rational number with denominator qn. Thus 1=qn ≤ jα − a=qj sup jp0(x)j: x2(α−1,α+1) This simple theorem immediately shows that Lioville's number is transcen- dental because it is approximated by a rational number far too well to be al- gebraic.
    [Show full text]
  • Arxiv:1701.06384V4 [Math.CO] 22 Nov 2017
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bern Open Repository and Information System (BORIS) ALGEBRAIC MATROIDS AND FROBENIUS FLOCKS GUUS P. BOLLEN, JAN DRAISMA, AND RUDI PENDAVINGH Abstract. We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindstr¨omvaluation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindstr¨omvaluation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations. 1. Introduction Several years before Whitney defined matroids in [23], Van der Waerden [22] recognized the commonality between algebraic and linear independence which is the defining property of matroids (see [18, Section 39.10b]). Today, linear matroids are the subject of some of the deepest theorems in combinatorics. By contrast, our understanding of algebraic matroids is much less advanced. Algebraic matroids seem forebodingly hard to deal with, compared to linear matroids. Deciding if a matroid has a linear representation over a given field amounts to solving a system of polynomial equations over that field, which makes the linear representation problem decidable over algebraically closed fields (for the situation of representability over Q, see [21]). We do not know of any procedure to decide if a matroid has an algebraic representation; for example, it is an open problem if the Tic-Tac-Toe matroid, a matroid of rank 5 on 9 elements, is algebraic [8].
    [Show full text]
  • Algebraic Matroids in Action
    ALGEBRAIC MATROIDS IN ACTION ZVI ROSEN, JESSICA SIDMAN, AND LOUIS THERAN Abstract. In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each of these settings the fundamental problem is to determine the extent to which certain unknowns depend algebraically on given data. This has, in turn, led to a resurgence of interest in algebraic matroids, which are the combinatorial formalism for algebraic (in)dependence. We give a self-contained introduction to algebraic matroids together with examples highlighting their potential application. 1. Introduction. Linear independence is a concept that pervades mathematics and applications, but the corresponding notion of algebraic independence in its various guises is less well studied. As noted in the article of Brylawski and Kelly [6], between the 1930 and 1937 editions of the textbook Moderne Algebra [40], van der Waerden changed his treatment of algebraic independence in a field extension to emphasize how the theory exactly parallels what is true for linear independence in a vector space, showing the influence of Whitney's [45] introduction of of matroids in the intervening years. Though var der Waerden did not use the language of matroids, his observations are the foundation for the standard definition of an algebraic matroid. In this article, we focus on an equivalent definition in terms of polynomial ideals that is currently useful in applied algebraic geometry, providing explicit proofs for results that seem to be folklore. We highlight computational aspects that tie the 19th century notion of elimination via resultants to the axiomatization of independence from the early 20th century to current applications.
    [Show full text]
  • Research Statement Algebraic Matroids: Structure and Applications
    RESEARCH STATEMENT ALGEBRAIC MATROIDS: STRUCTURE AND APPLICATIONS ZVI ROSEN Algebraic matroids are combinatorial objects that can be extracted from geometric problems, describing the independence structure on the coordinates. In this proposal, algebraic matroids are used to analyze applied problems, and their structure is explored. We give historical background to this topic, then set forth four projects. 1. Introduction Matroids were introduced in the early 20th century as a way of uniting disparate notions of \independence" from across mathematics. Among these notions were linear independence of vectors and graphic independence { defined by acyclicity on the subgraph corresponding to a set of edges. Algebraic independence over a field k, defined by the non-existence of polynomial relations with coefficients in k among elements of a set, is another such notion. Van der Waerden first defined algebraic matroids [vdW91], and they were studied by MacLane in an early matroid paper [ML38]. They were largely neglected after then until the 1970's. Ingleton and Main showed the existence of non-algebraic matroids in [IM75], and Bernt Lindstr¨omtreated algebraic representability in several papers (e.g. [Lin83, Lin86, Lin88]), as did some others ([DL87, Gor88]). These objects deserved to be studied in their own right: instead of thinking of elements in a field extension, let us think of coordinates of the function field of a variety. Then, any geometric problem involving special coordinates (like varieties describing physical quantitites in the real world, or varieties with combinatorially-defined ideals) boils down to an algebraic matroid. For this reason, in the past few years alone, low-rank matrix completion [KTTU12], rigidity theory [KRT13], probability distributions [KR14] and chemical reaction networks [MRHB14] have all benefited from insights given through algebraic matroids.
    [Show full text]
  • IDEALWISE ALGEBRAIC INDEPENDENCE for ELEMENTS of the COMPLETION of a LOCAL DOMAIN William Heinzer
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Nebraska Cooperative Fish & Wildlife Research Nebraska Cooperative Fish & Wildlife Research Unit -- Staff ubP lications Unit Summer 1997 IDEALWISE ALGEBRAIC INDEPENDENCE FOR ELEMENTS OF THE COMPLETION OF A LOCAL DOMAIN William Heinzer Christel Rotthaus Sylvia Wiegand Follow this and additional works at: https://digitalcommons.unl.edu/ncfwrustaff Part of the Aquaculture and Fisheries Commons, Environmental Indicators and Impact Assessment Commons, Environmental Monitoring Commons, Natural Resource Economics Commons, Natural Resources and Conservation Commons, and the Water Resource Management Commons This Article is brought to you for free and open access by the Nebraska Cooperative Fish & Wildlife Research Unit at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Nebraska Cooperative Fish & Wildlife Research Unit -- Staff ubP lications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. ILLINOIS JOURNAL OF MATHEMATICS Volume 41, Number 2, Summer 1997 IDEALWISE ALGEBRAIC INDEPENDENCE FOR ELEMENTS OF THE COMPLETION OF A LOCAL DOMAIN WILLIAM HEINZER, CHRISTEL ROTITIAUS AND SYLVIA WIEGAND 1. Introduction Over the past forty years many examples in commutative algebra have been con- structed using the following principle: Let k be a field, let S k[xl Xn]x,xn) be a localized polynomial ring over k, and let a be an ideal in the completion S of S such that the associated prims of a are in the generic formal fiber of S; that is, p N S (0) for each p Ass(S/a):. Then S embeds in S/a, the fraction field Q(S) of S embeds in the fraction ring of S/a, and for certain choices of a, the intersection D Q(S) f3 (S/a) is a local Noetherian domain with completion D S/a.
    [Show full text]