Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.


Introduction
Polynomial functions on non-commutative algebras over commutative rings come in two flavors: one, induced by polynomials with scalar coefficients, the other, by polynomials with coefficients in the non-commutative algebra itself [1, 4-9, 11, 12].
For the algebra of upper triangular matrices over a commutative ring, we show how polynomial functions with matrix coefficients can be described in terms of polynomial functions with scalar coefficients. In particular, we express integervaued polynomials with matrix coefficients in terms of integer-valued polynomials with scalar coefficients. The latter have been studied extensively by Evrard, Fares, Johnson [1] and Peruginelli [6] and have been characterized as polynomials that are integer-valued together with a certain kind of divided differences.
Also, our results have a bearing on several open questions in the subject of polynomial functions on non-commutative algebras. They allow us to answer in the affirmative, in the case of algebras of upper triangular matrices, two questions (see section 5) posed by N. Werner [12] : whether the set of null-polynomials on a non-commutative algebra forms a two-sided ideal and whether the set of integervalued polynomials on a non-commutative algebra forms a ring. In the absence of a substitution homomorphism for polynomial functions on non-commutative rings, neither of these properties is a given.
Also, our results on polynomials on upper triangular matrices show that we may be able to describe polynomial functions induced by polynomials with matrix coefficients by polynomial functions induced by polynomials with scalar coefficients, even when the relationship is not as simple as in the case of the full matrix algebra.
Let D be a domain with quotient field K, A a finitely generated torsion-free D-algebra and B = A ⊗ D K. To exclude pathological cases we stipulate that A ∩ K = D. Then the set of right integer-valued polynomials on A is defined as where f (a) is defined by substitution on the right of the coefficients, f (a) = k b k a k , for a ∈ A and f (x) = k b k x k ∈ B[x]. Left integer-valued polynomials are defined analogously, using left-substitution: | ∀a ∈ A f (a) ℓ ∈ A}, where f (a) ℓ = k a k b k . Also, we have integer-valued polynomials on A with scalar coefficients: Peruginelli and Werner [7] have characterized the algebras for which this relationship holds. We now take this investigation one step further and show for algebras of upper triangular matrices, where A = T n (D) and B = T n (K), that Int B (A) can be described quite nicely in terms of Int K (A) (cf. Remark 1.6, Theorem 4.2), but that the connection is not as simple as merely tensoring with A.
For a ring R, and n ≥ 1, let M n (R) denote the ring of n×n matrices with entries in R and T n (R) the subring of upper triangular matrices, i.e., the ring consisting of n × n matrices C = (c ij ) with c ij ∈ R and c ij = 0 =⇒ j ≥ i. Remark 1.1. Let R be a commutative ring. We will make extensive use of the ring isomorphism between the ring of polynomials in one variable with coefficients in M n (R) and the ring of n × n matrices with entries in R[x]: The restriction of ϕ to (T n (R))[x] gives a ring isomorphism between (T n (R))[x] and T n (R[x]).
When we reinterpret f as an element of M n (R[x]) via the ring isomorphism of Remark 1.1, we denote the (i, j)-th entry of f by f ij .
In other words, Also, we write [M] i j for the (i, j)-the entry of a matrix M. In particular, [f (C)] i j is the (i, j)-the entry of f (C), the result of substituting the matrix C for the variable (to the right of the coefficients) in f , and [f (C) ℓ ] i j the (i, j)-th entry of f (C) ℓ , the result of substituting C for the variable in f to the left of the coefficients.
We will work in a setting that allows us to consider integer-valued polynomials and null-polynomials on upper triangular matrices at the same time.
From now on, R is a commutative ring, S a subring of R, and I an ideal of S. Notation 1.3. Let R be commutative ring, S a subring of R and I an ideal of S.
Example 1.4. When D is a domain with quotient field K and we set R = K and S = I = D, then Int R (T n (S), T n (I)) = Int K (T n (D)) is the ring of integervalued polynomials on T n (D) with coefficients in K and Int Tn(R) (T n (S), T n (I)) = Int Tn(K) (T n (D)), the set of right integer-valued polynomials on T n (D) with coefficients in (T n (K)). We will show that the latter set is closed under multiplication and, therefore, a ring, cf. Theorem 5.4.
Example 1.5. When R is a commutative ring and we set S = R and I = (0), then Int R (T n (S), T n (I)) = N R (T n (R)) is the ring of those polynomials in R[x] that map every matrix in T n (R) to zero, and Int Tn(R) (T n (S), T n (I)) = N Tn(R) (T n (R)) is the set of polynomials in (T n (R))[x] that map every matrix in T n (R) to zero under right substitution. We will show that the latter set is actually an ideal of (T n (R))[x], cf. Theorem 5.2.
We illustrate here in matrix form our main result on the connection between the two kinds of polynomial functions on upper triangular matrices, those induced by polynomials with matrix coefficients on one hand, and those induced by polynomials with scalar coefficients on the other hand. (For details, see Theorem 4.2.) Remark 1.6. If we identify Int Tn(R) (T n (S), T n (I)) and Int ℓ Tn(R) (T n (S), T n (I))a priori subsets of (T n (R))[x] -with their respective images in T n (R[x]) under the ring isomorphism

path polynomials and polynomials with scalar coefficients
We will use the combinatorial interpretation of the (i, j)-th entry in the k-th power of a matrix as a weighted sum of paths from i to j.
Consider a set V together with a subset E of V × V . We may regard the pair (V, E) as a set with a binary relation or as a directed graph. Choosing the interpretation as a graph, we may associate monomials to paths and polynomials to finite sets of paths.
For our purposes, a path of length We introduce a set of independent variables with coefficients in the commutative ring R.
To each edge e = (a, b) in E, we associate the variable x ab and to each path e 1 e 2 . . . e k of length k, with e i = (a i , b i ), the monomial of degree k which is the product of the variables associated to the edges of the path: To each path of length 0 we associate the monomial 1.
If E is finite, or, more generally, if for any pair of vertices a, b ∈ V and fixed k ≥ 0 there are only finitely many paths in (V, E) from a to b, we define the k-th path polynomial from a to b, denoted by p (k) ab , as the sum in R[X] of the monomials corresponding to paths of length k from a to b. If there is no path of length k from a to b in (V, E), we set p From now on, we fix the relation (N, ≤) and all path polynomials will refer to the graph of (N, ≤), where N = {1, 2, 3, . . .}, or one of its finite subgraphs given by intervals. The finite subgraph given by the interval Because of the transitivity of the relation "≤", a path in (N, ≤) from a to b involves only vertices in the interval [a, b]. The path polynomial p (k) ab , therefore, is the same whether we consider a, b as vertices in the graph (N, ≤), or any subgraph given by an interval So we may suppress all references to intervals and subgraphs and define: for i > j and all k: p For a, b ∈ N, we define the sequence of path polynomials from a to b as ij is the (i, j)-th entry of the k-th power of a generic upper triangular n × n matrix (with n ≥ i, j) whose (i, j)-th entry is x ij when i ≤ j and zero otherwise. Again, note that p 24 is the sequence of entries in position (2,4) in the powers G 0 , G, G 2 , G 3 , . . . of a generic n×n (with n ≥ 4) upper triangular matrix G = (g ij ) with g ij = x ij for i ≤ j and g ij = 0 otherwise.
In addition to right and left substitution of a matrix for the variable in a polynomial in R[x] or (T n (R))[x], we are going to use another way of plugging matrices into polynomials, namely, into polynomials in For this purpose, the matrix C = (c ij ) ∈ M n (R) is regarded as a vector of elements of R indexed by N × N, with c ij = 0 for i > n or j > n: and a matrix C = (c ij ) ∈ M n (R) we define p(C) as the result of substituting c ij for those x ij in p with i, j ≤ n and substituting 0 for all x kh with k > n or h > n.
To be able to describe the (i, j)-th entry in f (C), where f ∈ R[x], we need one more construction: For sequences of polynomials , at least one of which is finite, we define a scalar product p, q = i p i q i . Actually, we only need one special instance of this construction, that where one of the sequences is the sequence of coefficients of a polynomial in R[x] and the other a sequence of path polynomials from a to b.
(which we identify with the sequence of its coefficients), a, b ∈ N, and p ab = (p  ( (3) If the i-th row or the j-th column of C is zero then p  Proof. The R-algebra isomorphism with ψ(x hk ) = ψ(x h+m k+m ) and ψ(r) = r for all r ∈ R maps f, p ij to f, p i+m j+m .
Applying ψ amounts to a renaming of variables; it doesn't affect the image of the polynomial function resulting from substituting elements of S for the variables. (1) f ∈ Int R (T n (S), T n (I)) (2) ∀ 1 ≤ i ≤ j ≤ n f, p ij ∈ Int(S * , I) Proof. The (i, j)-th entry of f (C), for C ∈ T n (R), is f, p ij (C), by Remark 2.7 (2). If C varies through T n (S), then all variables occurring in f, p ij vary through S independently. This shows the equivalence of (1) and (2).
By Lemma 2.8, the image of f, p ij as the variables range through S depends only on f and j − i. This shows the equivalence of (2) and (3).

Remark 3.2. Note that, for f ∈ R[x]
and C ∈ T n (R), This is so because no variables other than x st with i ≤ s ≤ t ≤ j occur in p (k) ij , for any k.
We derive some technical, but useful, formulas for the (i, j)-th entry in f (C) and f (C) ℓ , respectively, where f ∈ (T n (R))[x] and C ∈ T n (R).
and also hj (C).
By Remark 3.2, we can replace C by C [h,j] , the matrix obtained from C by replacing all entries with row or column index outside the interval [h, j] by zeros. Therefore, This shows the formulas for right substitution. Now, if we substitute C for the variable of f to the left of the coefficients, hj .
Changing summation, we get hj p .
hj = 0 only if h ≤ j, and that p By Remark 3.2, we can replace C by C [i,h] , the matrix obtained from C by replacing all entries in rows and columns with index outside [i, h] by zeros. Therefore, [right:] The following are equivalent [left:] The following are equivalent Proof. For right substitution: (2 ⇒ 1) follows directly from (1 ⇒ 2) Induction on h from j down to i. Given m ∈ [i, j], we show (2) for h = m, assuming that the statement holds for all values h ∈ [i, j] with h > m. In the above formula from Lemma 3.3, we let C vary through T For left substitution: (2 ⇒ 1) follows directly from , for all C ∈ T n (R), and secondly, as far as the image of f hj , p ih is concerned, all possible values under substitution of elements of S for the variables are realized as C varies through T n (S), since no variables other than x ij with 1 ≤ i ≤ j ≤ n occur in this polynomial.

Results for polynomials with matrix coefficients
Considering how matrix multiplication works, it is no surprise that, for a fixed row-index i and f ∈ (T n (R))[x], whether the entries of the i-th row of f (C) are in I for every C ∈ T n (S), depends only on the i-th rows of the coefficients of f . Indeed, if f = k F k x k , and [B] i denotes the i-th row of a matrix B, then Likewise, for a fixed column index j, whether the entries of the j-th column of f (C) ℓ are in I for every C ∈ T n (S) depends only on the j-th columns of the coefficients of f = k F k x k : if [B] j denotes the j-th column of a matrix B, then Now we can formulate the precise criterion which the i-th rows, or j-th columns, of the coefficients of f ∈ (T n (R))[x] have to satisfy to guarantee that the entries of the i-th row of f (C), or the j-th column of f (C) ℓ , respectively, are in I for every C ∈ T n (S).
[left:] Let 1 ≤ j ≤ n. The following are equivalent (1) For every C ∈ T n (S), all entries of the j-th column of f (C) ℓ are in I.
Proof. For right substitution: by Lemma 3.4, (1-3) are equivalent conditions for the (i, j)-th entry of f (C) to be in I for all j with i ≤ j ≤ n, for every C ∈ T n (S).
(3⇒4) For each h with i ≤ h ≤ n, letting j vary from h to n shows that criterion (3) of Proposition 2.9 for f ih ∈ Int R (T n−h+1 (S), T n−h+1 (I)) is satisfied.
For left substitution: by Lemma 3.4, (1-3) are equivalent conditions for the (i, j)-th entry of f (C) ℓ to be in I for all i with 1 ≤ i ≤ j, for every C ∈ T n (S).
(3⇒4) For each h with 1 ≤ h ≤ j, letting i range from 1 to h shows that criterion (3) of Proposition 2.9 for f hj ∈ Int R (T h (S), T h (I)) is satisfied.
We are now ready to prove the promised characterization of polynomials with matrix coefficients in terms of polynomials with scalar coefficients: Then f ∈ Int Tn(R) (T n (S), T n (I)) ⇔ ∀i, j f ij ∈ Int R (T n−j+1 (S), T n−j+1 (I)) and f ∈ Int ℓ Tn(R) (T n (S), T n (I)) ⇔ ∀i, j f ij ∈ Int R (T i (S), T i (I)). Proof. The criterion for f ∈ Int Tn(R) (T n (S), T n (I)) is just Lemma 4.1 applied to each row-index i of the coefficients of f ∈ (T n (R))[x], and the criterion for f ∈ Int ℓ Tn(R) (T n (S), T n (I)) is Lemma 4.1 applied to each column index j of the coefficients of f .
With the above proof we have also shown Remark 1.6 from the introduction, which is the representation in matrix form of Theorem 4.2.

Applications to null-polynomials and integer-valued polynomials
Applying our findings to null-polynomials on upper triangular matrices, we can describe polynomials with coefficients in T n (R) that induce the zero-function on T n (R) in terms of polynomials with coefficients in R that induce the zero function on T n (R). As before, we denote substitution for the variable in a polynomial f = k b k x k in T n (R)[x], to the right or to the left of the coefficients, as Null-polynomials on matrix algebras occur naturally in two circumstances: in connection with null-ideals of matrices [2,10], and in connection with integervalued polynomials. For instance, if D is a domain with quotient field K and f ∈ (T n (K))[x], we may represent f as g/d with d ∈ D and g ∈ (T n (D))[x]. Then f is (right) integer-valued on T n (D) if and only if the residue class of g in T n (D/dD)[x] is a (right) null-polynomial on T n (D/dD) [3].
From Theorem 4.2 we derive the following corollary (see also Remark 1.6): This allows us to conclude: Theorem 5.2. Let R be a commutative ring. The set N Tn(R) (T n (R)) of right nullpolynomials on T n (R) with coefficients in (T n (R))[x], and the set N ℓ Tn(R) (T n (R)) of left null-polynomials on T n (R) with coefficients in T n (R), are ideals of (T n (R))[x].
Proof. Note that N R (T m (R)) ⊆ N R (T k (R)) for m ≥ k. Also, N R (T m (R))R[x] ⊆ N R (T m (R)) and R[x]N R (T m (R)) ⊆ N R (T m (R)) by substitution homomorphism for polynomials with coefficients in the commutative ring R. This observation together with matrix multiplication shows that the image of N Tn(R) ((T n (R)) in T n (R Int K (T n (D)) Int K (T n−1 (D)) . . . Int K (T 2 (D)) Int K (T 1 (D)) 0 Int K (T n−1 (D)) . . . Int K (T 2 (D)) Int K (T 1 (D)) . . .  Proof. Note that Int K (T m (D)) ⊆ Int K (T k (D)) for m ≥ k. Together with substitution homomorphism for polynomials with coefficients in K, this implies Int K (T i (D)) · Int K (T j (D)) ⊆ Int K (T min(i,j) (D)).
This observation together with matrix multiplication shows that the image of Int Tn(K) (T n (D)) in T n (K[x]) under the ring isomorphism of (T n (K))[x] and T n (K[x]) is a subring of T n (K[x]), and, likewise, the image of Int ℓ Tn(K) (T n (D)) is a subring of T n (K[x]).