to appear in Monatsh. Math. (2017)

POLYNOMIAL FUNCTIONS ON UPPER TRIANGULAR MATRIX ALGEBRAS

SOPHIE FRISCH

Abstract. 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.

1. 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.

2010 Mathematics Subject Classification. Primary 13B25: Secondary 13F20, 16A42, 15A24, 05A05, 11C08, 16P10 .

Key words and phrases. integer-valued polynomials, null polynomials, zero polynomials, polynomial functions, upper triangular matrices, matrix algebras, polynomials on non-commutative algebras, matrices over commutative rings.

The research leading to this publication was supported by the Austrian Science Fund FWF, grant P27816-N26.

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

Int_{B}(A) = {f ∈ B[x] | ∀a ∈ A f(a) ∈ A},

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:

Int_{B}^{ℓ}(A) = {f ∈ B[x] | ∀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:

Int_{K}(A) = {f ∈ K[x] | ∀a ∈ A f(a) ∈ A}.

For A = M_{n}(D) and B = M_{n}(K) we could show [4] that

Int_{B}(A) = Int_{K}(A) ⊗_{D} A.

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]:

φ: (M_{n}(R))[x] → M_{n}(R[x]), ∑_{k}(a_{ij}^{(k)})_{1≤i,j≤n} x^{k} ↦ (∑_{k} a_{ij}^{(k)}x^{k})_{1≤i,j≤n}

The restriction of φ to (T_{n}(R))[x] gives a ring isomorphism between (T_{n}(R))[x] and T_{n}(R[x]).

Notation 1.2. For f ∈ (M_{n}(R))[x], F_{k} denotes the k-th coefficient of f, and f_{ij}^{(k)} ∈ R the (i, j)-th entry in F_{k}. 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,

f = ∑_{k} F_{k}x^{k}, where F_{k} = (f_{ij}^{(k)})_{1≤i,j≤n}

and

φ(f) = (f_{ij})_{1≤i,j≤n}, where f_{ij} = ∑_{k} f_{ij}^{(k)}x^{k}

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.

Int_{R}(T_{n}(S), T_{n}(I)) = {f ∈ R[x] | ∀C ∈ T_{n}(S) : f(C) ∈ T_{n}(I)}
Int_{T_{n}(R)}(T_{n}(S), T_{n}(I)) = {f ∈ (T_{n}(R))[x] | ∀C ∈ T_{n}(S) : f(C) ∈ T_{n}(I)}
Int^{ℓ}_{T_{n}(R)}(T_{n}(S), T_{n}(I)) = {f ∈ (T_{n}(R))[x] | ∀C ∈ T_{n}(S) : f(C)_{ℓ} ∈ T_{n}(I)},

where

f(C) = ∑_{k} F_{k}C^{k}  and f(C)_{ℓ} = ∑_{k} C^{k}F_{k}  for f = ∑_{k} F_{k}x^{k}.

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_{T_{n}(R)}(T_{n}(S), T_{n}(I)) = Int_{T_{n}(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 ideal of those polynomials in R[x] that map every matrix in T_{n}(R) to zero, and Int_{T_{n}(R)}(T_{n}(S), T_{n}(I)) = N_{T_{n}(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_{T_{n}(R)}(T_{n}(S), T_{n}(I)) and Int^{ℓ}_{T_{n}(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

φ: (T_{n}(R))[x] → T_{n}(R[x]), ∑_{k}(f_{ij}^{(k)})_{1≤i,j≤n} x^{k} ↦ (∑_{k} f_{ij}^{(k)}x^{k})_{1≤i,j≤n}

then

(1) Int_{T_{n}(R)}(T_{n}(S), T_{n}(I)) =

Int_{R}(T_{n}(S), T_{n}(I)) Int_{R}(T_{n−1}(S), T_{n−1}(I)) … Int_{R}(T_{2}(S), T_{2}(I)) Int_{R}(S, I)
0  Int_{R}(T_{n−1}(S), T_{n−1}(I)) … Int_{R}(T_{2}(S), T_{2}(I)) Int_{R}(S, I)
...
0  0  … Int_{R}(T_{2}(S), T_{2}(I)) Int_{R}(S, I)
0  0  …  0  Int_{R}(S, I)

(2) Int^{ℓ}_{T_{n}(R)}(T_{n}(S), T_{n}(I)) =

Int_{R}(S, I)  Int_{R}(S, I)  …  Int_{R}(S, I)  Int_{R}(S, I)
0  Int_{R}(T_{2}(S), T_{2}(I)) … Int_{R}(T_{2}(S), T_{2}(I))  Int_{R}(T_{2}(S), T_{2}(I))
...
0  0  … Int_{R}(T_{n−1}(S), T_{n−1}(I)) Int_{R}(T_{n−1}(S), T_{n−1}(I))
0  0  …  0  Int_{R}(T_{n}(S), T_{n}(I))

2. 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 k ≥ 1 from a to b (where a, b ∈ V) in a directed graph (V, E) is a sequence e_{1}e_{2} … e_{k} of edges e_{i} = (a_{i}, b_{i}) ∈ E such that a_{1} = a, b_{k} = b and b_{j} = a_{j+1} for 1 ≤ j < k. Also, for each a ∈ V, there is a path of length 0 from a to a, which we denote by ε_{a}. (For a ≠ b there is no path of length 0 from a to b.)

We introduce a set of independent variables X = {x_{v w} | (v, w) ∈ V × V } and consider the polynomial ring

R[X] = R[{x_{v w} | (v, w) ∈ V × V }]

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: x_{a_{1}b_{1}}x_{a_{2}b_{2}} … x_{a_{k}b_{k}}. 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_{ab}^{(k)}, 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_{ab}^{(k)} = 0.

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

[i, j] = {k ∈ ℕ | i ≤ k ≤ j}

is the graph with set of vertices [i, j] and set of edges {(a, b) | i ≤ a ≤ b ≤ j}.

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_{ab}^{(k)}, therefore, is the same whether we consider a, b as vertices in the graph (ℕ, ≤), or any subgraph given by an interval [i, j] with a, b ∈ [i, j]. So we may suppress all references to intervals and subgraphs and define:

Definition 2.1. Let R be a commutative ring. The k-th path polynomial from i to j (corresponding to the relation (ℕ, ≤)) in R[x] is defined by

(1) For 1 ≤ i ≤ j and k > 0 by

p_{ij}^{(k)} = ∑_{i=i_{1}≤i_{2}≤...≤i_{k+1}=j} x_{i_{1}i_{2}}x_{i_{2}i_{3}} … x_{i_{k−1}i_{k}}x_{i_{k}i_{k+1}},

(2) for 1 ≤ i ≤ j and k = 0 by p_{ij}^{(0)} = δ_{ij},

(3) for i > j and all k: p_{ij}^{(k)} = 0.

For a, b ∈ ℕ, we define the sequence of path polynomials from a to b as

p_{ab} = (p_{ab}^{(k)})_{k≥0}.

Remark 2.2. Note that p_{ij}^{(k)} 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.

Example 2.3. The sequence of path polynomials from 2 to 4 is

p_{24} = (0, x_{24}, x_{22}x_{24} + x_{23}x_{34} + x_{24}x_{44}, …)

and

p_{24}^{(3)} = x_{22}x_{22}x_{24} + x_{22}x_{23}x_{34} + x_{22}x_{24}x_{44} + x_{23}x_{33}x_{34} + x_{23}x_{34}x_{44} + x_{24}x_{44}x_{44}

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 R[X] = R[{x_{ij} | i, j ∈ ℕ}]. 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:

Definition 2.4. For a polynomial p ∈ R[X] = R[{x_{ij} | i, 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 p = (p_{i})_{i≥0}, q = (q_{i})_{i≥0} in R[X], 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.

Definition 2.5. Given f = f_{1} + f_{1}x + … f_{m}x^{m} ∈ R[x] (which we identify with the sequence of its coefficients), a, b ∈ ℕ, and p_{ab} = (p_{ab}^{(k)})_{k=0}^{∞} the sequence of path polynomials from a to b as in Definition 2.1, we define

⟨f, p_{ab}⟩ = ∑_{k≥0} f_{k} p_{ab}^{(k)}.

Definition 2.6. For a polynomial p ∈ R[X] = R[{x_{ij} | i, j ∈ ℕ}] and S ⊆ R, we define the image p(S^{∗}) ⊆ R as the set of values of p as the variables occurring in p range through S independently. (The star in S^{∗} serves to remind us that the arguments of p are not elements of S, but k-tuples of elements of S for unspecified k.)

We define Int(S^{∗}, I) as the set of those polynomials in R[X] = R[{x_{ij} | i, j ∈ ℕ}] that take values in I whenever elements of S are substituted for the variables.

The notation Int(S^{∗}, I) is suggested by the convention that Int(S, I) consists of polynomials in one indeterminate mapping elements of S to elements of I and, for k ∈ ℕ, Int(S^{k}, I) consists of polynomials in k indeterminates mapping k-tuples of elements of S to I.

We summarize here the connection between path poynomials and the related constructions of Definitions 2.1, 2.4, 2.5 and 2.6 with entries of powers of matrices and entries of the image of a matrix under a polynomial function.

Remark 2.7. Let R be a commutative ring, C ∈ T_{n}(R), k ≥ 0, 1 ≤ i, j ≤ n, and p_{ij}^{(k)} the k-the path polynomial from i to j in R[x] as in Definition 2.1.

(1) [C^{k}]_{ij} = p_{ij}^{(k)}(C)

(2) For f ∈ R[x], [f(C)]_{ij} = ⟨f, p_{ij}⟩(C).

(3) If the i-th row or the j-th column of C is zero then p_{ij}^{(k)}(C) = 0, and for all f ∈ R[x], ⟨f, p_{ij}⟩(C) = 0.

(4) p_{ij}^{(k)}(S^{∗}) = {p_{ij}^{(k)}(C) | C ∈ T_{n}(S)}.

Proof. (1) and (2) follow immediately from Definitions 2.1, 2.4 and 2.5. Compare Remark 2.2. (3) follows from (2) and Definition 2.4, since every monomial occurring in p_{ij}^{(k)} involves a variable x_{im} for some m and a variable x_{hj} for some h. Also,

(4) follows from Definitions 2.1 and 2.4. □

Lemma 2.8. Let f ∈ R[x]. The image of ⟨f, p_{ij}⟩ under substitution of elements of S for the variables depends only on f and j − i, that is, for all i ≤ j and all m ∈ ℕ

⟨f, p_{ij}⟩(S^{∗}) = ⟨f, p_{i+m j+m}⟩(S^{∗})

Proof. The R-algebra isomorphism

ψ: R[{x_{hk} | i ≤ h ≤ k ≤ j}] → R[{x_{hk} | i + m ≤ h ≤ k ≤ j + m}]

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. □

Proposition 2.9. Let f ∈ R[x]. The following are equivalent

(1) f ∈ Int_{R}(T_{n}(S), T_{n}(I))

(2) ∀ 1 ≤ i ≤ j ≤ n ⟨f, p_{ij}⟩ ∈ Int(S^{∗}, I)

(3) ∀ 0 ≤ k ≤ n − 1 ∃i ∈ ℕ ⟨f, p_{i i+k}⟩ ∈ 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). □

3. Lemmata for polynomials with matrix coefficients

Notation 3.1. Given C ∈ M_{n}(R) and 1 ≤ h ≤ j ≤ n, let C^{[h,j]} be the matrix obtained from C by replacing all entries with row- or column-index outside the interval [h, j] = {m ∈ ℕ | h ≤ m ≤ j} by zeros; and for S ⊆ R, let

T_{n}^{[h,j]}(S) = {C^{[h,j]} | C ∈ T_{n}(S)}.

Remark 3.2. Note that, for f ∈ R[x] and C ∈ T_{n}(R),

[f(C)]_{ij} = ⟨f, p_{ij}⟩(C) = ⟨f, p_{ij}⟩(C^{[i,j]}) = [f(C^{[i,j]})]_{ij} .

This is so because no variables other than x_{st} with i ≤ s ≤ t ≤ j occur in p_{ij}^{(k)}, 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).

Lemma 3.3. Let f ∈ (T_{n}(R))[x] and C ∈ T_{n}(R) with notation as in 1.2 and in Definitions 2.1, 2.4, and 2.5. Then, for all 1 ≤ i ≤ j ≤ n, we have

(R)

[f(C)]_{i j} = ∑_{h∈[i,j]} [f_{ih}(C)]_{h j} = ∑_{h∈[i,j]} [f_{ih}(C^{[h,j]})]_{h j}

and also

[f(C)]_{i j} = ∑_{h∈[i,j]} ⟨f_{ih}, p_{hj}⟩ (C) = ∑_{h∈[i,j]} ⟨f_{ih}, p_{hj}⟩ (C^{[h,j]}).

(L)

[f(C)_{ℓ}]_{i j} = ∑_{h∈[i,j]} [f_{hj}(C)]_{i h} = ∑_{h∈[i,j]} [f_{hj}(C^{[i,h]})]_{i h}

and also

[f(C)_{ℓ}]_{i j} = ∑_{h∈[i,j]} ⟨f_{hj}, p_{ih}⟩ (C) = ∑_{h∈[i,j]} ⟨f_{hj}, p_{ih}⟩ (C^{[i,h]}).

Proof. Let d = deg f.

[f(C)]_{i j} = ∑_{k=0}^{d} [F_{k}C^{k}]_{i j} = ∑_{k=0}^{d} ∑_{h∈[1,n]} f_{ih}^{(k)} [C^{k}]_{h j} = ∑_{k=0}^{d} ∑_{h∈[1,n]} f_{ih}^{(k)}p_{hj}^{(k)}(C).

Changing summation, we get ∑_{h∈[1,n]} ∑_{k=0}^{d} f_{ih}^{(k)}p_{hj}^{(k)}(C).

Considering that f_{ih}^{(k)} ≠ 0 only if i ≤ h, and that p_{hj}^{(k)} ≠ 0 only if h ≤ j, we can restrict summation to h ∈ [i, j]. Consequently, [f(C)]_{i j} equals

∑_{h∈[i,j]} ∑_{k=0}^{d} f_{ih}^{(k)}p_{hj}^{(k)}(C) = ∑_{h∈[i,j]} ⟨f_{ih}, p_{hj}⟩ (C) = ∑_{h∈[i,j]} [f_{ih}(C)]_{h j} .

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,

[f(C)]_{i j} = ∑_{h∈[i,j]} ⟨f_{ih}, p_{hj}⟩ (C^{[h,j]}) = ∑_{h∈[i,j]} [f_{ih}(C^{[h,j]})]_{h j} .

This shows the formulas for right substitution.

Now, if we substitute C for the variable of f to the left of the coefficients,

[f(C)_{ℓ}]_{i j} = ∑_{k=0}^{d} [C^{k}F_{k}]_{i j} = ∑_{k=0}^{d} ∑_{h∈[1,n]} [C^{k}]_{i h} f_{hj}^{(k)} = ∑_{k=0}^{d} ∑_{h∈[1,n]} p_{ih}^{(k)}(C)f_{hj}^{(k)}.

Changing summation, we get

∑_{h∈[1,n]} ∑_{k=0}^{d} p_{ih}^{(k)}(C)f_{hj}^{(k)} = ∑_{h∈[1,n]} ∑_{k=0}^{d} f_{hj}^{(k)}p_{ih}^{(k)}(C).

Considering that f_{hj}^{(k)} ≠ 0 only if h ≤ j, and that p_{ih}^{(k)} ≠ 0 only if i ≤ h, we can restrict summation to h ∈ [i, j]. Consequently, [f(C)_{ℓ}]_{i j} equals

∑_{h∈[i,j]} ∑_{k=0}^{d} f_{hj}^{(k)}p_{ih}^{(k)}(C) = ∑_{h∈[i,j]} ⟨f_{hj}, p_{ih}⟩ (C) = ∑_{h∈[i,j]} [f_{hj}(C)]_{i h} .

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,

[f(C)_{ℓ}]_{i j} = ∑_{h∈[i,j]} ⟨f_{hj}, p_{ih}⟩ (C^{[i,h]}) = ∑_{h∈[i,j]} [f_{hj}(C^{[i,h]})]_{i h} .

□

Lemma 3.4. Let f ∈ (T_{n}(R))[x] with notation as in 1.2 and in Definitions 2.1, 2.4, and 2.5. Let 1 ≤ i ≤ j ≤ n.

[right:] The following are equivalent

(1) [f(C)]_{i j} ∈ I for all C ∈ T_{n}(S)
(2) [f_{ih}(C)]_{h j} ∈ I for all C ∈ T_{n}(S), for all h ∈ [i, j].
(3) ⟨f_{ih}, p_{hj}⟩ ∈ Int_{R}(S^{∗}, I) for all h ∈ [i, j].

[left:] The following are equivalent

(1) [f(C)_{ℓ}]_{i j} ∈ I for all C ∈ T_{n}(S)
(2) [f_{hj}(C)]_{i h} ∈ I for all C ∈ T_{n}(S), for all h ∈ [i, j].
(3) ⟨f_{hj}, p_{ih}⟩ ∈ Int_{R}(S^{∗}, I) for all h ∈ [i, j].

Proof. For right substitution: (2 ⇒ 1) follows directly from

[f(C)]_{i j} = ∑_{h∈[i,j]} [f_{ih}(C)]_{h j} ,

which is Lemma 3.3.

(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_{n}^{[m,j]}(S) (as in 3.1). For such a C, the summands [f_{ih}(C)]_{h j} with h < m are zero, by Remark 2.7 (3). The summands with h > m are in I, by induction hypothesis. Therefore [f(C)]_{i j} ∈ I for all C ∈ T_{n}(S) implies [f_{im}(C)]_{m j} ∈ I for all C ∈ T_{n}^{[m,j]}(S). Since [f_{im}(C)]_{m j} = [f_{im}(C^{[m,j]})]_{m j}, by Remark 3.2, the statement follows for all C ∈ T_{n}(S).

Finally, (3 ⇔ 2) because, firstly, [f_{ih}(C)]_{h j} = ⟨f_{ih}, p_{hj}⟩ (C), for all C ∈ T_{n}(R), and secondly, as far as the image of ⟨f_{ih}, p_{hj}⟩ is concerned, all possible values under substitution of elements from S for the variables are obtained as C varies through T_{n}(S), because no variables other that x_{ij} with 1 ≤ i ≤ j ≤ n occur in this polynomial.

For left substitution: (2 ⇒ 1) follows directly from

[f(C)_{ℓ}]_{i j} = ∑_{h∈[i,j]} [f_{hj}(C)]_{i h} ,

which is Lemma 3.3.

(1 ⇒ 2) Induction on h, from h = i to h = j. Given m ∈ [i, j], we show (2) for h = m under the hypothesis that it holds for all h ∈ [i, j] with h < m. In the above formula from Lemma 3.3, the summands corresponding to h < m are in I by induction hypothesis. If we let C vary through T_{n}^{[i,m]}(S) (as in 3.1), then, for such C, the summands [f_{hj}(C)]_{i h} = ⟨f_{hj}, p_{ih}⟩(C) corresponding to h > m are zero, by Remark 2.7 (3). Therefore [f_{mj}(C^{[i,m]})]_{i m} ∈ I for all C ∈ T_{n}(S). But [f_{mj}(C^{[i,m]})]_{i m} = [f_{mj}(C)]_{i m} for all C ∈ T_{n}(S), by Remark 3.2. Thus, (2) follows by induction.

Finally, (3 ⇔ 2) because, firstly, [f_{hj}(C)]_{i h} = ⟨f_{hj}, p_{ih}⟩ (C), 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. □

4. 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

[f(C)]_{i} = ∑_{k}[F_{k}]_{i}C^{k}.

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

[f(C)_{ℓ}]^{j} = ∑_{k} C^{k}[F_{k}]^{j}.

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).

Lemma 4.1. Let f ∈ (T_{n}(R))[x]. We use the notation of 1.2 and Definitions 2.1. 2.4, and 2.5. [right:] Let 1 ≤ i ≤ n. The following are equivalent

(1) For every C ∈ T_{n}(S), all entries of the i-th row of f(C) are in I.

(2) For all C ∈ T_{n}(S), for all h, j with i ≤ h ≤ j ≤ n, [f_{ih}(C)]_{h j} ∈ I.

(3) For all h, j with i ≤ h ≤ j ≤ n, ⟨f_{ih}, p_{hj}⟩ ∈ Int_{R}(S^{∗}, I).

(4) f_{ih} ∈ Int_{R}(T_{n−h+1}(S), T_{n−h+1}(I)) for h = i, …, n.

[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.

(2) For all C ∈ T_{n}(S), for all i, h with 1 ≤ i ≤ h ≤ j, [f_{hj}(C)]_{i h} ∈ I.

(3) For all i, h with 1 ≤ i ≤ h ≤ j, ⟨f_{hj}, p_{ih}⟩ ∈ Int_{R}(S^{∗}, I).

(4) f_{hj} ∈ Int_{R}(T_{h}(S), T_{h}(I)) for h = 1, …, j.

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.

(4⇒3) For each fixed h, f_{ih} satisfies, by criterion (2) of Proposition 2.9, in particular, ⟨f_{ih}, p_{1k}⟩ ∈ Int(S^{∗}, I) for all 1 ≤ k ≤ n − h + 1. By Lemma 2.8, ⟨f_{ih}, p_{1k}⟩(S^{∗}) equals ⟨f_{ih}, p_{h h+k−1}⟩(S^{∗}). Therefore ⟨f_{ih}, p_{hj}⟩ ∈ Int(S^{∗}, I) for all j with h ≤ j ≤ n.

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.

(4⇒3) For each h with 1 ≤ h ≤ j, applying criterion (2) of Proposition 2.9 to f_{hj} shows, in particular, ⟨f_{hj}, p_{ih}⟩ ∈ Int(S^{∗}, I) for all 1 ≤ i ≤ h. □

We are now ready to prove the promised characterization of polynomials with matrix coefficients in terms of polynomials with scalar coefficients:

Theorem 4.2. Let f ∈ (T_{n}(R))[x]. Let F_{k} = (f_{ij}^{(k)})_{1≤i,j≤n} denote the coefficient of x^{k} in f and set

f_{ij} = ∑_{k} f_{ij}^{(k)}x^{k}.

Then

f ∈ Int_{T_{n}(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}_{T_{n}(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_{T_{n}(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}_{T_{n}(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.

5. 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

f(C) = ∑_{k} b_{k}C^{k} and f(C)_{ℓ} = ∑_{k} C^{k}b_{k}

and define

N_{T_{n}(R)}(T_{n}(R)) = {f ∈ (T_{n}(R))[x] | ∀C ∈ T_{n}(R) f(C) = 0}
N_{T_{n}(R)}^{ℓ}(T_{n}(R)) = {f ∈ T_{n}(R)[x] | ∀C ∈ T_{n}(R) f(C)_{ℓ} = 0}
N_{R}(T_{k}(R)) = {f ∈ R[x] | ∀C ∈ T_{n}(R) f(C) = 0}

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):

Corollary 5.1. If we identify polynomials in (T_{n}(R))[x] with their images in T_{n}(R[x]) under the isomorphism of 1.1, then N_{T_{n}(R)}(T_{n}(R)) =

N_{R}(T_{n}(R)) N_{R}(T_{n−1}(R)) … N_{R}(T_{2}(R)) N_{R}(T_{1}(R))
0  N_{R}(T_{n−1}(R)) … N_{R}(T_{2}(R)) N_{R}(T_{1}(R))
...
0  0  … N_{R}(T_{2}(R)) N_{R}(T_{1}(R))
0  0  …  0  N_{R}(T_{1}(R))

and N_{T_{n}(R)}^{ℓ}(T_{n}(R)) =

N_{R}(T_{1}(R)) N_{R}(T_{1}(R)) … N_{R}(T_{1}(R)) N_{R}(T_{1}(R))
0  N_{R}(T_{2}(R)) … N_{R}(T_{2}(R)) N_{R}(T_{2}(R))
...
0  0  … N_{R}(T_{n−1}(R)) N_{R}(T_{n−1}(R))
0  0  …  0  N_{R}(T_{n}(R))

This allows us to conclude:

Theorem 5.2. Let R be a commutative ring. The set N_{T_{n}(R)}(T_{n}(R)) of right nullpolynomials on T_{n}(R) with coefficients in (T_{n}(R))[x], and the set N_{T_{n}(R)}^{ℓ}(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_{T_{n}(R)}(T_{n}(R)) in T_{n}(R[x]) under the ring isomorphism from (T_{n}(R))[x] to T_{n}(R[x]) is an ideal of T_{n}(R[x]), and likewise the image of N_{T_{n}(R)}^{ℓ}(T_{n}(R)) in T_{n}(R[x]). □

Now let D be a domain with quotient field K. Applying Theorem 4.2 to integervalued polynomials (see also Remark 1.6)

Int_{T_{n}(K)}(T_{n}(D)) = {f ∈ (T_{n}(K))[x] | ∀C ∈ T_{n}(D) f(C) ∈ T_{n}(D)}
Int^{ℓ}_{T_{n}(K)}(T_{n}(D)) = {f ∈ (T_{n}(K))[x] | ∀C ∈ T_{n}(D) f(C)_{ℓ} ∈ T_{n}(D)}
Int_{K}(T_{n}(D)) = {f ∈ K[x] | ∀C ∈ T_{n}(D) f(C) ∈ T_{n}(D)}

yields the following corollary:

Corollary 5.3. If we identify polynomials in (T_{n}(K))[x] with their images in T_{n}(K[x]) under the isomorphism of 1.1, then

Int_{T_{n}(K)}(T_{n}(D)) =

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))
...
0  0  … Int_{K}(T_{2}(D)) Int_{K}(T_{1}(D))
0  0  …  0  Int_{K}(T_{1}(D))

and Int^{ℓ}_{T_{n}(K)}(T_{n}(D)) =

Int_{K}(T_{1}(D)) Int_{K}(T_{1}(D)) … Int_{K}(T_{1}(D)) Int_{K}(T_{1}(D))
0  Int_{K}(T_{2}(D)) … Int_{K}(T_{2}(D)) Int_{K}(T_{2}(D))
...
0  0  … Int_{K}(T_{n−1}(D)) Int_{K}(T_{n−1}(D))
0  0  …  0  Int_{K}(T_{n}(D))

We can see the following:

Theorem 5.4. Let D be a domain with quotient field K. The set Int_{T_{n}(K)}(T_{n}(D)) of right integer-valued polynomials with coefficients in (T_{n}(K))[x] and the set Int^{ℓ}_{T_{n}(K)}(T_{n}(D)) of left integer-valued polynomials with coefficients in (T_{n}(K))[x] are subrings of (T_{n}(K))[x].

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_{T_{n}(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}_{T_{n}(K)}(T_{n}(D)) is a subring of T_{n}(K[x]). □

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Institut für Mathematik A, Technische Universität Graz, Kopernikusgasse 24, 8010 Graz, Austria

E-mail address: frisch@math.tugraz.at