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ON VANDERMONDE VARIETIES

RALF FR

OBERG AND BORIS SHAPIRO

Abstract.Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties, which we callVandermonde varieties. They are closely related to the varieties consist- ing of all linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e., when the dimension of a Vandermonde variety is the expected one, we present its free resolution, obtain its degree and the Hilbert series. Some interesting relations among Schur polynomials are derived. Many open problems and conjectures are posed.

1.Introduction

The results in the present paper come from an attempt to understand the fa- mous Skolem-Mahler-Lech theorem and its consequences. Let us brie y recall its formulation. A linear recurrence relation with constant coecients of orderkis an equation of the form u n+1un1+2un2++kunk= 0; nk(1) where the coecients (1;:::;k) are xed complex numbers andk6= 0. (Equation (1) is often referred to as a linear homogeneous dierence equation with constant coecients.)

The left-hand side of the equation

t k+1tk1+2tk2++k= 0 (2) is called thecharacteristic polynomialof recurrence (1). Denote the roots of (2) (listed with possible repetitions) byx1;:::;xk;and call them thecharacteristic rootsof (1). Notice that allxiare non-vanishing sincek6= 0. To obtain a concrete solution of (1) one has to prescribe additionally an initialk-tuple, (u0;:::;uk1), which can be chosen arbitrarily. Thenun; nk;are obtained by using relation (1). A solution of (1) is callednon-trivialif not all of its entries vanish. In case of all distinct characteristic roots a general solution of (1) can be given by u n=c1xn1+c2xn2+:::+ckxnk wherec1;:::;ckare arbitrary complex numbers. In the general case of multiple characteristic roots a similar formula can be found in e.g. [15]. An arbitrary solution of a linear homogeneous dierence (or dierential) equa- tion with constant coecients of orderkis called anexponential polynomial of orderk. One usually substitutesxi6= 0 bye iand considers the obtained func- tion inCinstead ofZorN. (Other terms used for exponential polynomials are quasipolynomialsorexponential sums.) The most fundamental fact about the structure of integer zeros of exponential polynomials is the well-known Skolem-Mahler-Lech theorem formulated below. It was rst proved for recurrence sequences of algebraic numbers by K. Mahler [11]

in the 30's, based upon an idea of T. Skolem [13]. Then, C. Lech [9] published2010Mathematics Subject Classication.Primary 65Q10, Secondary 65Q30, 14M15.

1

2 R. FR

OBERG AND B. SHAPIRO

the result for general recurrence sequences in 1953. In 1956 Mahler published the same result, apparently independently (but later realized to his chagrin that he had actually reviewed Lech's paper some years earlier, but had forgotten it). Theorem 1(The Skolem-Mahler-Lech theorem).Ifa0;a1;:::is a solution to a lin- ear recurrence relation, then the set of all k such thatak= 0is the union of a nite (possibly empty) set and a nite number (possibly zero) of full arithmetic progres- sions. (Here, a full arithmetic progression means a set of the formr;r+d;r+ 2d;::: with0< r < d:) A simple criterion guaranteeing the absence of arithmetic progressions is that no quotient of two distinct characteristic roots of the recurrence relation under consideration is a root of unity, see e.g. [10]. A recurrence relation (1) satisfying this condition is callednon-degenerate. Substantial literature is devoted to nding the upper/lower bounds for the maximal number of arithmetic progressions/exceptional roots among all/non-degenerate linear recurrences of a given order. We give more details inx3. Our study is directly inspired by these investigations. LetLkbe the space of all linear recurrence relations (1) of order at mostk with constant coecients. Denote byLk=Lkn fk= 0gthe subset of all linear recurrence of order exactlyk. (Lkis the ane space with coordinates (1;:::;k).) To an arbitrary pair (k;I) wherek2 is a positive integer andI=fi0< i1< i2< ::: < i m1g; mk;is a sequence of integers, we associate the varietyVk;ILk, the set of all linear recurrences of order exactlyk;having a non-trivial solution vanishing at all points ofI. Denote byV k;Ithe closure ofVk;IinLkin the usual topology. We callVk;I(resp.V k;I) the open (resp. closed)linear recurrence variety associated to the pair (k;I). In what follows we will always assume that gcd(i1i0;:::;im1i0) = 1 to avoid unnecessary freedom related to the time rescaling in (1). Notice that since formk1 one hasVk;I=LkandV k;I=Lk, this case does not require special consideration. A more important observation is that due to translation invariance of (1) for any integerland any pair (k;I) the varietyVk;I(resp.V k;I) coincides with the varietyVk;I+l(resp.V k;I+l) where the set of integersI+lis obtained by addinglto all entries ofI.

So far we denedV

k;IandVk;Ias sets. However for any pair (k;I) the setV k;Iis an ane algebraic variety, see Proposition 4. Notice that this fact is not completely obvious since if we, for example, instead of a set of integers choose asI an arbitrary subset of real or complex numbers then the similar subset ofLnwill, in general, only be analytic. Now we dene the Vandermonde variety associated with a given pair (k;I); I= f0i0< i1< i2< ::: < im1g; mk. Firstly, consider the setMk;Iof (generalized) Vandermonde matrices of the form M k;I=0 B B@x i01xi02xi0kxi11xi12xi1k x im11xim12xim1 k1 C

CA;(3)

where (x1;:::;xk)2Ck. In other words, for a given pair (k;I), we take the map M k;I:Ck!Mat(m;k) given by (3) whereMat(m;k) is the space of allmk- matrices with complex entries and (x1;:::;xk) are chosen coordinates inCk. We now dene three slightly dierent but closely related versions of this variety as follows.

ON VANDERMONDE VARIETIES 3

Version 1.Given a pair (k;I) withjIj k, dene thecoarse Vandermonde variety V d ck;IMk;Ias the set of all degenerate Vandermonde matrices, i.e., whose rank is smaller thank.V dck;Iis obviously an algebraic variety whose dening idealII is generated by allm kmaximal minors ofMk;I. Denote the quotient ring by R

I=R=II.

Denote byAkCkthe standard Coxeter arrangement (of the Coxeter group A k1) consisting of all diagonalsxi=xj, and byBCkCkthe Coxeter arrange- ment consisting of allxi=xjandxi= 0. Obviously,BCk Ak. Notice that V d ck;Ialways includes the arrangementBCkifi0>0 (some of the hyperplanes with multiplicities), which is often inconvenient. Namely, with very few exceptions this means thatV dck;Iis not equidimensional, not CM, not reduced etc. For applications to linear recurrences as well as questions in combinatorics and geometry of Schur polynomials it seems more natural to consider the localizations ofV dck;IinCknAk and inCkn BCk. Version 2.Dene theAk-localizationV dAk;IofV dck;Ias the contraction ofV dck;I toCkn Ak. Its is easy to obtain the generating ideal ofV dAk;I. Namely, recall that given a sequenceJ= (j1< j2<< jk) of nonnegative integers, one denes the associated Schur polynomialSJ(x1;:::;xk) as given by S

J(x1;:::;xk) =

x j11xj12xj1k xj21xj22xj2k x jk1xjk2xjkk =W(x1;:::;xk); whereW(x1;:::;xk) is the usual Vandermonde determinant. Given a sequence I= (0i0< i1< i2<< im1) with gcd(i1i0;:::;im1i0) = 1, consider the set of all itsm ksubsequencesJof lengthk. Here the indexruns over the set of all subsequences of lengthkamongf1;2;:::;mg. Take the corresponding Schur polynomialsSJ(x1;:::;xk) and form the idealIAIC[x1;:::;xk] generated by allm ksuch Schur polynomialsSJ(x1;:::;xk). One can show that the Vandermonde varietyV dAk;ICkis dened byIAIsettheoretically;see Lemma 5. Denote the quotient ring byRAI=R=IAIwhereR=C[x1;:::;xk]. Analogously, to the coarse Vandermonde varietyV dAk;Ioften contains irrelevant coordinate hyperplanes which prevents it from having nice algebraic properties. For example, ifi0>0 then all coordinate hyperplanes necessarily belong toV dAk;Iruining equidimensionality etc. On the other hand, under the assumption thati0= 0 the varietyV dAk;Ioften has quite reasonable properties presented below. Version 3.Dene theBCk-localizationV dBCk;IofV dck;Ias the contraction ofV dck;I toCkn BCk. Again it is straightforward to nd the generating ideal ofV dBCk;I. Namely, given a sequenceJ= (0j1< j2<< jk) of nonnegative integers dene the reduced Schur polynomial ^SJ(x1;:::;xk) as given by

SJ(x1;:::;xk) =

1 11 x j2j11xj2j12xj2j1k x jkj11xjkj12xjkj1k =W(x1;:::;xk):

In other words,

^SJ(x1;:::;xk) is the usual Schur polynomial corresponding to the sequence (0;j2j1;::::;jkj1). Given a sequenceI= (0i0< i1< i2<< i m1) with gcd(i1i0;:::;im1i0) = 1, consider as before the set of all itsm k subsequencesJof lengthkwhere the indexruns over the set of all subsequences of lengthk. Take the corresponding reduced Schur polynomials^SJ(x1;:::;xk) and

4 R. FR

OBERG AND B. SHAPIRO

form the idealIBCIC[x1;:::;xk] generated by allm ksuch Schur polynomials ^SJ(x1;:::;xk). One can easily see that the Vandermonde varietyV dBCk;ICkis dened set-theoretically byIBCI:Denote the quotient ring byRBCI=R=IBCI. Conjecture 2.If dim(V dBCk;I)2 thenIBCIis a radical ideal. Notice that considered as sets the restrictions toCkn BCkof all three varieties V d ck;I,V dAk;I,V dBCk;Icoincide with what we call theopen Vandermonde variety V d op k;Iwhich is the subset of all matrices of the formMk;Iwith three properties: (i) rank is smaller thank; (ii) allxi's are non-vanishing; (iii) allxi's are pairwise distinct. Thus set-theoretically all the dierences between the three Vandermonde vari- eties are concentrated on the hyperplane arrangementBCk. Also from the above denitions it is obvious thatV dop k;IandV dBCk;Iare invariant under addition of an arbitrary integer toI. The relation between the linear recurrence varietyVk;Iand the open Vandermonde varietyV dop k;Iis quite straight-forward. Namely, consider the standard Vieta map:

V i:Ck!Lk(4)

sending an arbitraryk-tuple (x1;:::;xk) to the polynomialtk+1tk1+2tk2++ kwhose roots arex1;::::;xk. Inverse images of the Vieta map are exactly the orbits of the standardSk-action onCkby permutations of coordinates. Thus, the Vieta map sends a homogeneous and symmetric polynomial to a weighted homogeneous polynomial.

Dene theopen linear recurrence varietyVop

k;IVk;Iof a pair (k;I) as consist- ing of all recurrences inVk;Iwith all characteristic roots distinct. The following statement is obvious.

Lemma 3.The mapV irestricted toV dop

k;Igives an unramiedk!-covering of the setVop k;I. Unfortunately at the present moment the following natural question is still open.

Problem 1.Is it true that thatV

op k;I=Vk;Ifor any pair (k;I), whereV op k;Iis the set- theoretic closure ofVop k;IinLk? If 'not', then under what additional assumptions? Our main results are as follows. Using the Eagon-Northcott resolution of de- terminantal ideals, we determine the resolution, and hence the Hilbert series and degree ofRAIin Theorem 6. We give an alternative calculation of this degree using the Giambelli-Thom-Porteous formula in Proposition 8. In the simplest non-trivial case, whenm=k+ 1, we get more detailed information aboutV dAk;I. We prove that its codimension is 2, and thatRAIis Cohen-Macaulay. We also discuss minimal sets of generators ofII, and determine when we have a complete intersection in Theorem 9. (The proof of this theorem gives some interesting relations between Schur polynomials, see Theorem 10.) In this case the variety has the expected codimension, which is not always the case form > k+ 1. In fact our computer experiments suggest that then the codimension rather seldom is the expected one. In casek= 3,m= 5, we show that having the expected codimension is equivalent toRAIbeing a complete intersection, and thatIIis generated by three complete symmetric functions. Exactly the problem (along with many other similar ques- tions) when three complete symmetric functions constitute a regular sequence was considered in a paper [4], where the authors formulated a detailed conjecture. We slightly strengthen their conjecture below.

ON VANDERMONDE VARIETIES 5

For theBCk-localized varietyV dBCk;Iwe have only proofs whenk= 3, but we present Conjectures 15 and 16, supported by many calculations. We end the paper with a section which describes the connection of our work with the fundamental problems in linear recurrence relations. Acknowledgements.The authors want to thank Professor Maxim Kazarian (Steklov Institute of Mathematical Sciences) for his help with Giambelli-Thom-Porteous for- mula, Professor Igor Shparlinski (Macquarie University) for highly relevant infor- mation on the Skolem-Mahler-Lech theorem and Professors Nicolai Vorobjov (Uni- versity of Bath) and Michael Shapiro (Michigan State University) for discussions. We are especially grateful to Professor Winfried Bruns (University of Osnabruck) for pointing out important information on determinantal ideals. We also thank the anonymous referee for a careful reading, and especially for noting a mistake in

Theorem 6.

2.Results and conjectures on Vandermonde varieties

We start by proving thatV

k;Iis an ane algebraic variety, see Introduction.

Proposition 4.For any pair(k;I)the setV

k;Iis an ane algebraic variety.

Therefore,Vk;I=V

k;IjLkis a quasi-ane variety. Proof.We will show that for any pair (k;I) the varietyV k;Iof linear recurrences is constructible. Since it is by denition closed in the usual topology ofLk'Ck, it is algebraic. The latter fact follows from [12], I.10 Corollary 1, claiming that ifZXis a constructible subset of a variety, then the Zarisky closure and the strong closure ofZare the same. Instead of showing thatV k;Iis constructible, we prove thatVk;ILkis constructible. Namely, we can use an analog of Lemma 3 to construct a natural stratication ofVk;Iinto the images of quasi-ane sets under appropriate Vieta maps. Namely, let us stratifyVk;IasVk;I=S `kVk;I where`kis an arbitrary partition ofkandVk;Iis the subset ofVk;Iconsisting of all recurrence relations of lengthkwhich has a non-trivial solution vanishing at each point ofIand whose characteristic polynomial determines the partition of its degreek. In other words, if= (1;:::;s);Ps j=1j=kthen the characteristic polynomial should havesdistinct roots of multiplicities1;:::;s resp. Notice that any of theseVk;Ican be empty including the wholeVk;Iin which case there is nothing to prove. Let us now show that eachVk;Iis the image under the appropriate Vieta map of a set similar to the open Vandermonde variety. Recall that if= (1;:::;s);Ps j=1j=kandx1;::;xsare the distinct roots with the multiplicities1;:::;srespectively of the linear recurrence (1) then the general solution of (1) has the form u n=P1(n)xn1+P2(n)xn2+:::+Ps(n)xns; whereP1(n);:::;Ps(n) are arbitrary polynomials in the variablenof degrees1

1;21;::::;s11 resp. Now, for a given`kconsider the set of matrices

M k;I= 0 B B@x i01i0xi01::: i11

0xi01::: xi0si0xi0s::: is1

0xi0s xi11i1xi11::: i11

1xi11::: xi1si1xi1s::: is1

1xi1s

x im11im1xim11::: i11 m1xim11::: xim1si1xim1s::: is1 m1xim1s1 C CA: In other words, we are taking the fundamental solutionxn1;nxn1;:::;n11xn1; xn2;quotesdbs_dbs50.pdfusesText_50

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