On the rational relationships among pseudo-roots of a non-commutative polynomial
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 where t commutes with all elements of R. Since t is central, every product with is equal to a monomial of the form , and every element of has a normal form . As usual, for , the evaluation is defined to be , and α is a zero of p if . Some familiar properties of polynomials over commutative rings fail for non-commutative rings. The identity 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 over quaternions has infinitely many roots.
Following [20], [6], an element is a pseudo-root of p provided there exist polynomials such that . We call ξ a right root of p if , and a left root of p if . It is easy to verify that ξ is a zero of p if and only if ξ is a right-root of p. Let be the set of zeros of p and be the set of pseudo-roots of p.
If the polynomial factors as , then are pseudo-roots, and is a zero. For a commutative domain R, of course, every permutation of the factors is a factorization, and . 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 is a left principal ideal ring and the set of monic polynomials is in 1-1 correspondence with the left ideals. Thus with the divisibility order is a lattice with join operation (least common multiple) equal to the unique monic generator of the left ideal , and meet operation (greatest common divisor) equal to the unique monic generator of .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 conjugacy classes of R and if p factors as then each zero of p is conjugate to some .
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 as rational functions of zeros 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 that are “suitably generic”, there are unique elements and such that . Call this polynomial . Then are all pseudo-roots of with being zeros. We can express and as rational functions of via the formulas and can express as rational functions of via the formulas provided that the needed inverses are well-defined (this is what is meant by the above requirement that be “suitably generic”). However, one cannot (in general) rationally express either or in terms of [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 is a right divisor of r, denoted , if there is a polynomial s such that . The polynomial s is unique, and is denoted by . Let be the directed graph on vertex set , the set of monic polynomials in , whose arc-set consists of pairs for all such that and . Every arc can be naturally associated to an element of R.
Let be the restriction of to the set of right divisors of p, and let be the arc-set of . In the case of present interest where R is a division ring and p is factorizable, is the cover graph of the lattice of divisors of p, i.e., for any divisors q and r of p, if and only if there is a directed path from r to q in . It is easy to check from the definitions that for every , is a pseudo-root of p, and every pseudo-root is representable in this form for some (not necessarily unique) arc of .
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 is finite, then the set of polynomials that vanish at every is a left ideal and so is generated by a unique monic polynomial, which we denote by . Polynomials of the form were studied by Lam and Leroy [16], who called them Wedderburn polynomials. In the case that R is a field, is just the product . As shown by Lam and Leroy (see [16]) for any subset S, the set is closed under gcd and lcm3 and is thus a sublattice of . We denote this sublattice by . Note that is a finite sublattice of the lattice , and has at most elements (polynomials). The sublattices of the form give a rich source of examples of finite sublattices of .
Let be the subgraph of consisting only of the vertices in and the arcs between them and write for the set of arcs of . Then, as above, each arc corresponds to a pseudo-root of . Let denote the set of pseudo-roots of corresponding to the arc set . We refer to as S-pseudo-roots. It is easy to see that if then every directed path from the maximal element to the minimal element 1, consists of a sequence of polynomials where . If is the pseudo-root associated to the arc then the product is a factorization of . In this way every path from to 1, corresponds to a factorization of into linear factors where each factor is t minus a pseudo-root.
For a subset B of R let 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 of S-pseudo-roots. More precisely, for , let , i.e., the set of S-pseudo-roots that are rationally generated by B. The map is a closure map on (see Section 2.1), and our goal is to provide a partial description of this map.
Under the correspondence between arcs of and S-pseudo-roots , the closure operator on can be interpreted as a closure operator on the set of arcs of : For , , where .
While depends on the ring-theoretic structure of S within R, it was observed in [6] that can be partially captured by a natural closure on the arc-set , introduced in work of I. Gelfand and Retakh [9], [10], that depends only on the graph structure of . 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 are vertices in such that and and , then the set is a diamond of . The diamond closure of , denoted is the smallest set of arcs containing B that satisfies that for any diamond if contains or then contains the entire diamond. This definition is motivated by Example 1.1, where a is the polynomial , b is , c is and d is 1, and arc corresponds to pseudo-root , corresponds to pseudo-root , corresponds to and corresponds to . The analysis of Example 1.1 can be extended (see Proposition 2.4) to show that for any ring R and finite , every member of 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 for . 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 for any subset B of arcs. In particular, this provides a sufficient condition for a set of pseudo-roots in to rationally generate all of . 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 to rationally generate .
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 consisting of a set X and a closure map that satisfies (1) , (2) implies and (3) . The image of λ is denoted . Members of are λ-closed subsets. If C is λ-closed and satisfies 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 for all . 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 is its cover set.
We note the following easy observation: Proposition 3.1 A set is diamond-closed if and only if each arc-component of S is diamond-closed.
The set 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 .
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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