On the rational relationships among pseudo-roots of a non-commutative polynomial

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Abstract

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t)=p0+p1t+pktk is an element ξR, for which there exist polynomials q1,q2 such that p=q1(tξ)q2. We investigate the rational relationships that hold among the pseudo-roots of p(t) by using the diamond operations for cover graphs of modular lattices.

When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial fS that vanishes on S. By results of Leroy and Lam [16], the set of polynomials {fT:TS} with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set ΛS of pseudo-roots of fS. Given an arbitrary subset of ΛS, our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of ΛS, and in particular, rationally generates S.

Introduction

The theory of polynomials with noncommutative coefficients and central variables was initiated by Wedderburn, Dickson and Ore (see, e.g., [15], Chapter 5.16 and [19]). There is a significant literature on polynomials with matrix coefficients and their factorizations into linear factors, for example, [3], [12], [18]. For factorizations of noncommutative polynomials in a more general setting see, for example [16], [17].

Let R be an arbitrary ring, and consider the ring R[t] where t commutes with all elements of R. Since t is central, every product a1ar with aiR{t} is equal to a monomial of the form atd, and every element of R[t] has a normal form p=p0+p1t++pktk. As usual, for αR, the evaluation p(α) is defined to be p0+p1α++pkαk, and α is a zero of p if p(α)=0. Some familiar properties of polynomials over commutative rings fail for non-commutative rings. The identity pq(α)=p(α)q(α) need not hold. While every zero of q is a zero of pq, a zero of p need not be a zero of pq. A degree d polynomial may have more than d distinct zeros, e.g., the polynomial t2+1 over quaternions has infinitely many roots.

Following [20], [6], an element ξR is a pseudo-root of p provided there exist polynomials q1,q2R[t] such that p=q1(tξ)q2. We call ξ a right root of p if q2=1, and a left root of p if q1=1. It is easy to verify that ξ is a zero of p if and only if ξ is a right-root of p. Let Z(p) be the set of zeros of p and Λ(p) be the set of pseudo-roots of p.

If the polynomial p(t) factors as (tα1)(tαd), then α1,,αd are pseudo-roots, and αd is a zero. For a commutative domain R, of course, every permutation of the factors is a factorization, and Z(p)={α1,,αd}. In the non-commutative case, a permutation of a factorization need not be a factorization, and a polynomial may have many factorizations that are not equivalent under permutation.

Throughout the paper, we assume that R is a division ring. Therefore R[t] is a left principal ideal ring and the set of monic polynomials is in 1-1 correspondence with the left ideals. Thus R[t] with the divisibility order is a lattice L(R) with join operation lcm(p1,p2) (least common multiple) equal to the unique monic generator of the left ideal R[t]p1R[t]p2, and meet operation gcd(p1,p2) (greatest common divisor) equal to the unique monic generator of R[t]p1+R[t]p2.2 This lattice is necessarily modular (see Section 2.5).

Connections among pseudo-roots for polynomials over division rings are given by the famous Gordon-Motzkin theorem (see [13] or Chapter 5.16 in [15]): The zeros of any polynomial p lie in at most deg(p) conjugacy classes of R and if p factors as (tα1)(tαd) then each zero of p is conjugate to some αi.

Exact conjugation formulas connecting zeros and pseudo-roots over division rings were given by Gelfand and Retakh in [9] (see also [10], [5]). They expressed coefficients of polynomial p=tn+a1tn1++an as rational functions of zeros ξ1,,ξn provided that the roots are in generic position (see Section 2.7).

The following simple example from [9] is instructive.

Example 1.1

Given elements x1,x2R that are “suitably generic”, there are unique elements x1 and x2 such that (tx1)(tx1)=(tx2)(tx2). Call this polynomial p(t). Then x1,x2,x1,x2 are all pseudo-roots of p(t) with x1,x2 being zeros. We can express x1 and x2 as rational functions of x1,x2 via the formulasx1=(x1x2)x1(x1x2)1,x2=(x2x1)x2(x2x1)1, and can express x1,x2 as rational functions of x1,x2 via the formulasx1=(x1x2)1x1(x1x2),x2=(x2x1)1x2(x2x1), provided that the needed inverses are well-defined (this is what is meant by the above requirement that x1,x2 be “suitably generic”). However, one cannot (in general) rationally express either x2 or x2 in terms of x1,x1 [1].

This example suggests the following general problem: given a set B of pseudo-roots of a polynomial p and another pseudo-root α is α rationally generated by B? This is the focus of the present paper.

To formalize our problem, we need a way to specify individual pseudo-roots of a polynomial. As we now describe, there is a natural directed graph whose edges correspond to pseudo-roots of p. For monic polynomials q,r, q is a right divisor of r, denoted q|r, if there is a polynomial s such that r=sq. The polynomial s is unique, and is denoted by r/q. Let G=G(R) be the directed graph on vertex set RM[t], the set of monic polynomials in R[t], whose arc-set A(R) consists of pairs rq for all q,r such that q|r and deg(q)=deg(r)1. Every arc rq can be naturally associated to an element ψ(rq)=tr/q of R.

Let Gp=Gp(R) be the restriction of G(R) to the set

of right divisors of p, and let Ap be the arc-set of Gp. In the case of present interest where R is a division ring and p is factorizable, Gp is the cover graph of the lattice of divisors of p, i.e., for any divisors q and r of p, q|r if and only if there is a directed path from r to q in Gp. It is easy to check from the definitions that for every rqAp, ψ(rq) is a pseudo-root of p, and every pseudo-root is representable in this form for some (not necessarily unique) arc of Ap.

The lattice

is, in general, infinite and so is the set of pseudo-roots of p, and one gains greater control over the problem by restricting to certain natural finite sublattices, that were studied by Lam and Leroy [16]. If SR is finite, then the set of polynomials that vanish at every sS is a left ideal and so is generated by a unique monic polynomial, which we denote by fS. Polynomials of the form fS were studied by Lam and Leroy [16], who called them Wedderburn polynomials. In the case that R is a field, fS is just the product sS(ts). As shown by Lam and Leroy (see [16]) for any subset S, the set {fT:TS} is closed under gcd and lcm3 and is thus a sublattice of L(R). We denote this sublattice by LS. Note that LS is a finite sublattice of the lattice
, and has at most 2|S| elements (polynomials). The sublattices of the form LS give a rich source of examples of finite sublattices of L(R).

Let GS be the subgraph of GfS consisting only of the vertices in LS and the arcs between them and write AS for the set of arcs of GS. Then, as above, each arc rq corresponds to a pseudo-root ψ(rq) of fS. Let ΛS denote the set of pseudo-roots of fS corresponding to the arc set AS. We refer to ΛS as S-pseudo-roots. It is easy to see that if deg(fS)=k then every directed path from the maximal element fS to the minimal element 1, consists of a sequence fk=fS,fk1,,f0=1 of polynomials where deg(fj)=j. If aj is the pseudo-root associated to the arc fjfj1 then the product (tak)(tak1)(ta1) is a factorization of fS. In this way every path from fS to 1, corresponds to a factorization of fS into linear factors where each factor is t minus a pseudo-root.

For a subset B of R let Φ(B) denote the closure of B under ring operations and inversion of units. For a subset S we want to consider the restriction of this closure to the set ΛS of S-pseudo-roots. More precisely, for BΛS, let ΦS(B)=Φ(B)ΛS, i.e., the set of S-pseudo-roots that are rationally generated by B. The map ΦS:P(ΛS)P(ΛS) is a closure map on ΛS (see Section 2.1), and our goal is to provide a partial description of this map.

Under the correspondence between arcs of AS and S-pseudo-roots ΛS, the closure operator ΦS on Λ(p) can be interpreted as a closure operator on the set AS of arcs of GS: For BAS, ΦS(B)={rqAS:ψ(rq)Φ(ψ(B)), where ψ(B)={ψ(rq):rqB}.

While ΦS depends on the ring-theoretic structure of S within R, it was observed in [6] that ΦS can be partially captured by a natural closure on the arc-set AS, introduced in work of I. Gelfand and Retakh [9], [10], that depends only on the graph structure of GS. This closure, called the diamond-closure, and denoted by ⋄ can be described briefly as follows (see Section 2.3 for a more precise definition). If {a,b,c,d} are vertices in GS such that ab,ac and bd and cd, then the set {ab,ac,bd,cd} is a diamond of GS. The diamond closure of BAS, denoted (B) is the smallest set of arcs containing B that satisfies that for any diamond {ab,ac,bd,cd} if (B) contains {ab,ac} or {bd,cd} then (B) contains the entire diamond. This definition is motivated by Example 1.1, where a is the polynomial p(t), b is tx1, c is tx2 and d is 1, and arc ab corresponds to pseudo-root x1, ac corresponds to pseudo-root x2, bd corresponds to x1 and cd corresponds to x2. The analysis of Example 1.1 can be extended (see Proposition 2.4) to show that for any ring R and finite BR, every member of (B) is an S-pseudo-root that is rationally generated by B.

In this paper we investigate the ⋄-closure for finite graphs G that are cover graphs of modular lattices. This includes the motivating situation that the graph G is equal to GS for SR. We give an explicit description of ⋄-closed sets (Theorem 3.3) and use this to describe a simple procedure (see Theorem 3.11) for determining (B) for any subset B of arcs. In particular, this provides a sufficient condition for a set of pseudo-roots in ΛS to rationally generate all of ΛS. In the special case that the lattice is distributive, we can use the Birkhoff representation for distributive lattices to provide a more explicit characterization of connected diamond-closed sets (Theorem 4.5) which yields a simpler sufficient condition for a set of pseudo-roots of ΛS to rationally generate ΛS.

Our methods can be also applied to more general situations including skew-polynomials (see [16]), linear differential operators [11] and principal ideal rings [4].

Section snippets

Closure spaces

A closure space is a pair (X,λ) consisting of a set X and a closure map λ:2X2X that satisfies (1) Aλ(A), (2) AB implies λ(A)λ(B) and (3) λ(λ(A))=λ(A). The image {A:λ(A)=A} of λ is denoted Cλ. Members of Cλ are λ-closed subsets. If C is λ-closed and AC satisfies λ(A)=C we say that A is a λ-generating set for C.

If λ and μ are closure operators on the same set X, we say that λ is weaker than μ provided that λ(A)μ(A) for all AX. This is equivalent to the condition that every μ-closed set is

Diamond-closure in a finite modular lattice

In this section we provide a simple description of the diamond-closed sets of a modular lattice of finite length. Throughout this section L denotes an arbitrary finite modular lattice and CL is its cover set.

We note the following easy observation:

Proposition 3.1

A set SCL is diamond-closed if and only if each arc-component of S is diamond-closed.

The set (S) can be constructed from S by the following direct procedure: If there is any diamond D spanned by S that is not contained in S then replace S by SD.

Diamond-closure in distributive lattices

Theorem 3.11 describes the diamond-closure operation for a finite modular lattice. In this section we give a simplified description in the case that L is a finite distributive lattice.

We first use the Birkhoff representation theorem for distributive lattices to give a convenient way to describe the sublattices and the cover-preserving sublattices of a distributive lattice.

Acknowledgements

We thank an anonymous referee for carefully reading the paper and for identifying technical issues that have been corrected for this final version.

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M.S. is supported in part by Simons Foundation under grant 332622.

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