Báo cáo toán học: "The degree of a q-holonomic sequence is a quadratic quasi-polynomial" docx

The degree of a q-holonomic sequenceis a quadratic quasi-polynomial Stavros Garoufalidis ∗ School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160, USA stavros@math.

The degree of a q-holonomic sequence

is a quadratic quasi-polynomial

Stavros Garoufalidis ∗ School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160, USA stavros@math.gatech.edu

http://www.math.gatech.edu/∼stavros

Submitted: Jun 30, 2010; Accepted: Mar 5, 2011; Published: Mar 15, 2011

Mathematics Subject Classification: 05C88

Abstract

A sequence of rational functions in a variable q is q-holonomic if it satisfies

a linear recursion with coefficients polynomials in q and qn We prove that the degree of a q-holonomic sequence is eventually a quadratic quasi-polynomial, and that the leading term satisfies a linear recursion relation with constant coefficients Our proof uses differential Galois theory (adapting proofs regarding holonomic D-modules to the case of q-holonomic D-D-modules) combined with the Lech-Mahler-Skolem theorem from number theory En route, we use the Newton polygon of a linear q-difference equation, and introduce the notion of regular-singular q-difference equation and a WKB basis of solutions of a linear q-difference equation at q = 0 We then use the Skolem-Mahler-Lech theorem to study the vanishing of their leading term Unlike the case of q = 1, there are no analytic problems regarding convergence

of the WKB solutions Our proofs are constructive, and they are illustrated by an explicit example

Contents

1.1 History 2

1.2 The degree and the leading term of a q-holonomic sequence 3

1.3 The Newton polygon of a linear q-difference equation 4

1.4 WKB sums 5

1.5 An example 7

1.6 Plan of the paper 10

∗ To Doron Zeilberger, on the occasion of his 60th birthday

2 Proof of Theorem 1.2 11

2.1 Reduction to the case of a single slope 11

2.2 Reduction to the case of a single eigenvalue 12

2.3 First order linear q-difference equation 13

2.4 Proof of Theorem 1.2 14

2.5 The regular-singular non-resonant case 14

3 Proof of Theorem 1.4 16 3.1 Generalized power sums 16

3.2 Proof of theorem 1.4 17

4 Invariants of q-holonomic sequences 18 4.1 Synopsis of invariants 18

4.2 The annihilating polynomial of a q-holonomic sequence 18

4.3 The characteristic variety of a q-holonomic sequence 19

4.4 The Newton polytope of a q-holonomic sequence 19

4.5 The tropical curve of a q-holonomic sequence 20

4.6 A tropical equation for the degree of a q-holonomic sequence 20

4.7 Future directions 21

q-holonomic sequences appear in abundance in Enumerative Combinatorics; [PWZ96,

theorem of Wilf-Zeilberger states that a multi-dimensional finite sum of a (proper) q-hyper-geometric term is always q-holonomic; see [WZ92, Zei90, PWZ96] Given this result, one can easily generate q-holonomic sequences We learnt about this astonishing result from Doron Zeilberger in 2002 Putting this together with the fact that many state-sum invariants in Quantum Topology are multi-dimensional sums of the above shape, it follows that Quantum Topology provides us with a plethora of q-holonomic sequences

of natural origin; [GL05] For example, the sequence of Jones polynomials of a knot and its parallels (technically, the colored Jones function) is q-holonomic Moreover, the corresponding minimal recursion relation can be chosen canonically and is conjecturally related to geometric invariants of the knot; see [Gar04] Recently, the author focused on the degree of a q-holonomic sequence, and in the case of the Jones polynomial it is also conjecturally related to topological invariants of knot complement; see [Gar11] Since little is known about the Jones polynomial of a knot and its parallels, one might expect

no regularity on its sequence of degree The contrary is true, and in fact is a property of q-holonomic sequences and the focus of this paper Our results were announced in [Gar11] and [Gar], where numerous examples of geometric/topological origin were discussed

1.2 The degree and the leading term of a q-holonomic sequence

Our main theorem concerns the degree and the leading term of a q-holonomic sequence

To phrase it, we need to recall what is a q-holonomic sequence, and what is a polynomial A sequence (fn(q)) of rational functions is q-holonomic if it satisfies a recur-sion relation of the form

quasi-ad(qn, q)fn+d(q) + · · · + a0(qn, q)fn(q) = 0 (1)for all n where aj(u, v) ∈ Q[u, v] for j = 0, , d and ad(u, v) 6= 0

Given a polynomial with rational coefficients

K = Q{{q}} = ∪∞

r=1Q((q1/r)) (2)

of all Puiseux series in q with algebraic coefficients (see [Wal78]) The field K is required

in Theorems 1.2 and 1.3 below

Recall that a quasi-polynomial p(n) is a function

Theorem 1.1 follows from Theorems 1.2 and 1.4 below

Remark 1.1 If fn(q) ∈ Z[q±1] is q-holonomic with degree δn and leading term ltn, itfollows that fn(q) − ltnqδn is also q-holonomic Thus, Theorem 1.1 applies to each one ofthe terms of fn(q)

Remark 1.2 L Di Vizio brought to our attention that results similar to part (a) ofTheorem 1.1 appear in [BB92, Thm.4.1] and also in [DV08, Sec.1.1,Sec.1.3] However,the statement and proof of Theorem 1.1 appear to be new.

Our next corollary to Theorem 1.1 characterizes which sequences of monomials areq-holonomic sequences In a sense, such sequences are building blocks of all q-holonomicsequences

Corollary 1.3 If (an) and (bn) are sequences of integers (with an 6= 0 for all n), then(anqb n) is q-holonomic if and only if bnis holonomic and anis a quadratic quasi-polynomialfor all but finitely many n

1.3 The Newton polygon of a linear q-difference equation

As is common, we can write a linear q-difference equation (1) in operator form by ducing two operators L and M which act on a sequence (fn(q)) by

We will call an element of D a linear q-difference operator The general element of D is

of its monomials Since we are working locally at q = 0, we view N′(P ) by placing oureye at −∞ in the vertical axis and looking up The resulting object is the lower convexhull N(P ) defined above The Newton polygon of a linear q-difference equation was alsostudied in the recent Ph.D thesis by P Horn; see [Hor09, Chpt.2]

The Newton polygon N(P ) of P is a finite union of intervals with rational end pointsand two vertical rays Each interval with end-points (i, di) and (j, dj) for i < j has a

slope s = (dj − di)/(j − i) and a length (or multiplicity) l = j − i The multiset of slopess(N(P )) of N(P ) is the set of slopes of s N(P ) each with multiplicity ls An example of

a Newton polygon of a linear q-difference operator of degree 7 with three slopes −1, 0, 1/2

of multiplicity 2, 1, 4 respectively is shown here:

The Newton polygon is a convenient way to organize solutions to a linear q-differenceequation The reader may compare this with the Newton polygon of a linear differentialoperator attributed to Malgrange and Ramis; see [vdPS03, Sec.3.3] and references therein

In analogy with the theory of linear differential operators, we will say that P is a singularq-difference operator if its Newton polygon consists of a single horizontal segment

regular-In other words, after a minor change of variables, with the notation of (5) this meansthat ai(M, q) ∈ K[[M1/r]] for all i and a0(0, q)ad(0, q) 6= 0

below

Definition 1.4 (a) A formal WKB series is an expression of the form

fτ(u, q) = qγn 2

λ(q)nA(n, u, q) (7)where τ = (γ, λ(q), A) and

• the exponent γ is a rational number,

(b) A formal WKB sum is a finite K-linear combination of formal WKB series.

Observe that if fτ(u, q) is a formal WKB series, then its evaluation

fτ,n(q) = fτ(qn, q) ∈ K (8)

is a well-defined K-valued sequence for n > −rc

Observe further if fτ(u, q) is given by (7), the operators L and M act on fτ(u, q) by:(Lfτ)(u, q) = qγ(n+1)2λ(q)n+1A(n + 1, uq, q), (Mfτ)(u, q) = qγn2λ(q)nuA(n, u, q) (9)Theorem 1.2 Every linear q-difference equation has a basis of solutions of the form

fτ,n(q) Moreover, the multiset of exponents of the bases is the multiset of the negatives

of the slopes of the Newton polygon

Recall that K{{M}} denotes the field of Puiseux series in a variable M with coefficients

in K Let δM(f ) denote the minimum exponent of M in a Puiseux series f ∈ K{{M}}.Theorem 1.3 Every monic q-difference operator P ∈ D of order d can be factored as

where ai(M, q) ∈ K{{M}} and the multiset {δM(ai)|i = 1, , d} of slopes is the negative

of the multiset of slopes of the Newton polygon of P

Remark 1.5 The WKB expansion of solution of linear q-difference equations given inTheorem 1.2 appears to be new, and perhaps it is related to some recent work of Witten[Wit], who proposes a categorification of the colored Jones polynomial in an arbitrary3-manifold The curious reader may compare [Wit, Eqn.6.21] with our Theorem 1.2 Wewish to thank T Dimofte for pointing out the reference to us

Remark 1.6 Although the proofs of Theorems 1.2 and 1.3 follow the well-studied case

of linear differential operators in one variable x, we are unable to formulate an analogue

of Theorems 1.2 and 1.3 in the differential operator case Indeed, q is a variable whichseems to be independent of the spacial variables x of a differential operator The meaning

of the variable q can be explained by Quantization, or from Tropical Geometry (where it

is usually denoted by t) or from the representation theory of Quantum Groups; see forexample [Gar] and referencies therein, and also Section 4below

Theorem 1.4 Iffn(q) is a finite K-linear combination of formal WKB series of the form(7), then for large n its degree is given by a quadratic quasi-polynomial and its leadingterm satisfies a linear recursion relation with constant coefficients

The 2-dimensional Newton polytope N2(Pf) and the Newton polygon N (Pf) are given by:

Pf is a second order regular-singular q-difference operator Its Newton polygon N (Pf) has onlyone slope 0 with multiplicity 2 The edge polynomial of the 0-slope is cyclotomic (L − 1)2 withtwo equal roots (i.e., eigenvalues) 1 and 1 The degree δn and the leading term ltn of fn(q) aregiven by δn= 0 and ltn= n + 1 for all n

The characteristic curve chf (discussed in Section4) is reducible given by the zeros (L, M ) ∈(C∗)2 of the polynomial

(−1 + M + M2)(−1 + 2L − L2+ L2M − 3LM2+ LM3+ LM4) = 0

The tropical.lib program of [Mar] computes the vertices of the tropical curve Tf are:

(−1, −2), (3, −2), (0, −3/2), (1, −3/2), (0, −1)The next figure is a drawing of the tropical curve Tf and its multiplicities, where edges notlabeled have multiplicity 1

2

22

22

The Newton subdivision of the Newton polytope N2(Pf) is:

To illustrate Theorem 1.1, observe that fn(q) is a sequence of Laurent polynomials Theminimum (resp., maximum) degree δn (resp., ˆδn) of fn(q) with respect to q is given by:

δn= 0, δˆn= n(3n + 1)

2The coefficient ltn (resp., bltn) of qδn (resp., qˆn) in fn(q) is given by:

+q

2+k(1 + q) 1 + qk

ψ−5+k(−1 + qk) −q

1+k 1 + q + q2

1 + qk

ψ−4+k(−1 + qk)

−2q

−2+k q3+ q4− qk+ q3+k+ q4+k

ψ−3+k(−1 + qk)

+q

−2+k 2q2+ q3+ 2q4− 2q6− 2qk+ 2q1+k+ 2q2+k+ q3+k+ 2q4+k

ψ−2+k(−1 + qk)

+q

−1+k q + q2− 2q3− 2qk− q1+k+ q2+k

ψ−1+k(−1 + qk)

)

with an arbitrary initial condition (ψ0, φ0) ∈ Q[[q]]2 Note that ψk the first equation above is

a recursion relation for ψk and the second equation is a recursion for φk that also involves ψk′

for k′ < k This is a general feature of the WKB in the case of resonanse It follows that ourparticular solution (11) of the q-difference equation Pff = 0 has the form:

The reader may recognize (and also confirm by an explicit computation) that

for all n, much like the case of the colored Jones polynomial of a knot, [GL05]

1.6 Plan of the paper

Theorem1.1 follows from Theorems1.2and 1.4

Theorem 1.2 follows by two reductions using a q-analogue of Hensel’s lemma, analogous tothe case of linear differential operators The first reduction factors an operator with arbitrary

Newton polygon into a product of operators with a Newton polygon with a single slope After

a change of variables, we can assume that these operators are regular-singular The secondreduction factors a regular-singular q-difference operator into a product of first order regular-singular operators with eigenvalues of possibly equal constant term Thus, we may assume thatthe q-difference operator is regular-singular with eigenvalues of equal constant term In thatcase, we can construct a basis of formal WKB solutions of the required type The above proofalso factorizes a linear q-difference operator into a product of first order q-difference operators,giving a proof of Theorem 1.3

Theorem 1.4 follows from the Lech-Mahler-Skolem theorem on the zeros of generalized ponential sums

ex-Finally, in Section 4, which is independent of the results of the paper, we discuss severalinvariants of q-holonomic sequences, and compute them all in some simple examples

2.1 Reduction to the case of a single slope

Theorem1.2follows ideas similar to the case of linear differential operators, presented for ple in [vdPS03, Sec.3] Rather than develop the whole theory, we will highlight the differencesbetween the differential case and the q-difference case

exam-In the case of linear differential operators, the basic operators x and x∂x satisfy the mogeneous commutation relation

The right hand side of the above commutation relation leads to consider Newton polygons areconvex hulls of translates of the second quadrant R−× R+ For examples of such polygons, see[vdPS03, Sec.3.3] As a result, the slopes of the Newton polygons of linear differential operatorsare always non-negative rational numbers On the other hand, in the q-difference case, theq-commutation relation (3) preserves the L and M degree of a monomial in M, L

In the case of linear differential operators of a single variable x, the WKB solutions involvepower series in x1/r and polynomials in log x In the case of linear q-difference operators, theWKB solutions involve power series in qn and polynomials in n In addition, the convergenceconditions are easier to deal with when q = 0, and much harder when q = 1 Convergence when

q = 1 involves regularizing factorially divergent formal power series

Consider a q-difference operator P and its Newton polygon N (P ) In this section, we willallow the coefficients ai(M, q) of P from Equation (5) to be arbitrary elements of the fieldK{{M }} Recall that the Newton polygon N (P ) consists of finitely many segments and tworays We will say that P is slim if all its monomials lie in the bounded segments of its Newtonpolygon We will say that P1 > P2 if the boundary of N (P1) lies in the interior of P2 or in theinterior of the two rays of P2 We will say that an operator P from Equation (5) is monic if

ad(M, q) = 1 Recall the Minkowski sum

N1+ N2 = {a + b| a ∈ N1, b ∈ N2}

of two polytopes in the plane Here and below, we will always consider the Minkowski sum oftwo convex polytopes that arise from linear q-difference operators, i.e., polytopes that consist oftwo vertical rays and some bounded segments