Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Abstract

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and $1 < n_1 \le \cdots \le n_k$, there exists an integer-valued polynomial on D, that is, an element of ${{\,\textrm{Int}\,}}(D) = \{ f \in K[X] \mid f(D) \subseteq D \}$, which has precisely k essentially different factorizations into irreducible elements of ${{\,\textrm{Int}\,}}(D)$ whose lengths are exactly $n_1, \ldots , n_k$. Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

1 Introduction

In a foregoing manuscript [6], Frisch and the present authors investigated factorizations of integer-valued polynomials on valuation rings V of global fields K. They showed that the factorization behaviour in this case is as wild as can be: For all positive integers k and all integers $1 < n_1 \le \cdots \le n_k$, there exists an integer-valued polynomial H of

$$\begin{aligned}{{\,\textrm{Int}\,}}(V) = \{f \in K[X] \mid f(V) \subseteq V \}\end{aligned}$$

which has precisely k essentially different factorizations into irreducible elements of ${{\,\textrm{Int}\,}}(V)$ whose lengths are exactly $n_1,\ldots ,n_k$. This result gave a partial answer to the following open problem:

Problem

([4, Problem 39]). Analyze and describe non-unique factorization in ${{\,\textrm{Int}\,}}(V)$, where V is a DVR with finite residue field.

The above problem stands in a long tradition of the study of factorizations in integer-valued polynomial rings. Building on results by Cahen and Chabert [2] and Chapman and McClain [5], Frisch [7] showed that every finite multiset of integers $>1$ occurs as the set of lengths (of factorizations into irreducibles) of some polynomial in ${{\,\textrm{Int}\,}}(\mathbb {Z})$. This was generalized to rings of integer-valued polynomials on Dedekind domains with infinitely many maximal ideals of finite index by Frisch, Nakato, and Rissner [9] and is analogous to the result on DVRs above [6].

Since, so far, rings of integer-valued polynomials are not accessible by a general unified theory of factorizations, as it is the case for Krull domains, one has been forced to take a closer look to special multiplicative properties of particular (irreducible) elements of these rings.

First steps into the direction of such a unified theory of factorizations for integer-valued polynomials were made by Reinhart [12] and Frisch [8], respectively, who proved that ${{\,\textrm{Int}\,}}(D)$ is monadically Krull whenever D is a unique factorization domain (UFD), respectively, more generally, a Krull domain. This means that the local multiplicative behaviour in such rings works as in Krull domains. In a follow-up paper, Reinhart [13] then determined class groups of the monadic submonoids of ${{\,\textrm{Int}\,}}(D)$. However, in practice, these class groups are very hard to compute and therefore have not led to concrete applications.

A special situation occurs, when considering ${{\,\textrm{Int}\,}}(D)$, where D is a Krull domain all of whose height-one prime ideals have infinite index. In this case, ${{\,\textrm{Int}\,}}(D)=D[X]$, so ${{\,\textrm{Int}\,}}(D)$ is a Krull domain with divisor class group $\mathcal {C}_v({{\,\textrm{Int}\,}}(D)) \cong \mathcal {C}_v(D)$ and prime divisors in all classes. This implies that the factorization behavior of ${{\,\textrm{Int}\,}}(D)$ is determined by the combinatorial object $\mathcal {B}(\mathcal {C}_v(D))$, the monoid of zero-sum sequences over the class group. This is a very well investigated object and two cases are to distinguish: 1. $\mathcal {C}_v(D)$ is infinite: Then ${{\,\textrm{Int}\,}}(D)$ has full system of sets of lengths (arbitrary multisets of lengths need not be realizable). 2. $\mathcal {C}_v(D)$ is finite: For this case, a vast amount of literature is available and a lot is known.

In contrast to the effort investigating factorizations in rings of integer-valued polynomials, interestingly even the local case, that is, the case of integer-valued polynomials on DVRs, remained widely open. In the present manuscript, we close this gap and, in particular, solve the open problem above. In fact, we prove a stronger result:

Theorem

Let D be a Krull domain admitting a prime element $\pi$ such that $D/\pi D$ is finite. Then ${{\,\textrm{Int}\,}}(D)$ has full system of multisets of lengths, i.e., for all positive integers k and all integers $1 < n_1 \le \cdots \le n_k$, there exists an integer-valued polynomial $H \in {{\,\textrm{Int}\,}}(D)$ which has precisely k essentially different factorizations into irreducible elements of ${{\,\textrm{Int}\,}}(D)$ whose lengths are exactly $n_1,\ldots ,n_k$.

Note that the class of Krull domains admitting a principal prime ideal includes the following:

• (1) Unique factorization domains (UFDs) so, in particular,

• (2) Discrete valuation rings,

• (3) Rings of integers in global fields,

• (4) Monoid algebras that are Krull domains, which include polynomial rings over Krull domains.

In the case of UFDs, one can observe a particularly nice dichotomy: Depending on the existence of a principal prime ideal (or, equivalently, height-one prime ideal) with finite residue field, factorizations of elements are either unique or as wild as possible.

Corollary

Let D be a unique factorization domain. Exactly one of the following holds:

• (1) ${{\,\textrm{Int}\,}}(D) = D[X]$ is a unique factorization domain.

• (2) ${{\,\textrm{Int}\,}}(D)$ has full system of multisets of lengths, i.e., for all positive integers k and all integers $1 < n_1 \le \cdots \le n_k$, there exists an integer-valued polynomial $H \in {{\,\textrm{Int}\,}}(D)$ which has precisely k essentially different factorizations into irreducible elements of ${{\,\textrm{Int}\,}}(D)$ whose lengths are exactly $n_1,\ldots ,n_k$.

Moreover, (1) holds if and only if the residue field of each height-one prime ideal in D is infinite.

2 Preliminaries

Factorizations. We give an informal presentation of factorizations. The interested reader is refered to the monograph by Geroldinger and Halter–Koch [10] for a systematic introduction.

Let R be an integral domain and $r\in R$. We say that r is irreducible (in R) if it is a non-zero non-unit and cannot be written as the product of two non-units of R. A factorization of r is a decomposition

$$\begin{aligned}r=a_1\cdots a_n\end{aligned}$$

into irreducible elements $a_i$ of R. In this case, n is called the length of this factorization of r. Let s be a further element of R. We say that r and s are associated if there exists a unit $\varepsilon \in R$ such that $r=\varepsilon s$. We want to consider factorizations up to order and associates. In other words, two factorizations

$$\begin{aligned}r=a_1\cdots a_n=u_1\cdots u_m\end{aligned}$$

of r are essentially the same if $n=m$ and, after re-indexing if necessary, $u_i$ is associated to $a_i$ for all $i\in \{1,\ldots ,n\}$. Otherwise, the factorizations are called essentially different.

Valuations. Let K be a field. A valuation $\mathsf v$ on K is a map

$$\begin{aligned}\mathsf v:K^\times \rightarrow G\end{aligned}$$

where $(G,+,\le )$ is a totally ordered Abelian group, subject to the following conditions for all $a,b\in K^\times$:

• (1) $\mathsf v(a\cdot b)=\mathsf v(a) +\mathsf v(b)$ and

• (2) $\mathsf v(a+b)\ge \inf \{\mathsf v(a),\mathsf v(b)\}$.

The set $\{0\}\cup \{x\in K^\times \mid \mathsf v(x)\ge 0\}$ is called the valuation ring of $\mathsf v$. It is a subring of K with quotient field K.

We will often use implicitly the following fact about valuations, which follows from the definition by an easy exercise: If $\mathsf v$ is a valuation on K and $a,b\in K$ are such that $\mathsf v(a)\ne \mathsf v(b)$, then

$$\begin{aligned}\mathsf v(a+b)= \inf \{\mathsf v(a),\mathsf v(b)\}.\end{aligned}$$

If $\mathsf v(K^\times )\cong \mathbb {Z}$, we call $\mathsf v$ a discrete valuation (by the more precise terminology of Bourbaki it would be a discrete rank one valuation). If $\mathsf v$ is a discrete valuation on K, then there exists a valuation $\mathsf w:K^\times \rightarrow \mathbb {Z}$ with $\mathsf w(K^\times )=\mathbb {Z}$ and the same valuation ring as $\mathsf v$. We call $\mathsf w$ the normalized valuation of this valuation ring.

For a general introduction to valuations, see [1].

Newton polygons. Let $(K,\mathsf v)$ be a discretely valued field, where $\mathsf v$ is normalized, and let $f=a_0+a_1X+\cdots +a_nX^n$ be a polynomial over K. The Newton polygon of f is contructed as follows. To every term $a_iX^i$ we associate the point $(i,\mathsf v(a_i))\in \mathbb {Z}^2$, where we just ignore a point if its value in the second coordinate is infinity. We now form the lower boundary of the convex hull of the set of points $\{(0,\mathsf v(a_0)),\ldots ,(n,\mathsf v(a_n))\}$ and call it the Newton polygon of f. For an introduction to this topic, see [11, Ch. II, §6].

Integer-valued polynomials. Let R be an integral domain with quotient field K. The set

$$\begin{aligned}\text {Int}(R)=\{f\in K[X]\mid f(R)\subseteq R\}\end{aligned}$$

is a subring of K[X] and called the ring of integer-valued polynomials on R. Let V be the valuation ring of a valuation $\mathsf v$ on a field K. Every element $f\in K[X]$ can be written in the form $f=\frac{g}{d}$, where $g\in V[X]$ and $d\in V\setminus \{0\}$. It is immediate that $f\in {{\,\textrm{Int}\,}}(V)$ if and only if $\min _{a\in V} \mathsf v(f(a))\ge \mathsf v(d)$.

For a detailed treatment of integer-valued polynomials, we refer to the monograph by Cahen and Chabert [3].

We need the following fact that seems to be folklore. Lacking a proper reference, we give a proof of the statement.

Remark 2.1

Let D be an integral domain and $P \subseteq D$ be a prime ideal. Let $F \in D[X] \setminus P[X]$ be a polynomial.

If $\deg F < |D/P|$, then $F(a) \notin P$ for some $a \in D$.

Proof

Assume to the contrary that $F(a) \in P$ for all $a \in D$. Then the reduction $\overline{F}$ of F modulo P is a non-zero polynomial that has a zero at every element of D/P. This is a contradiction to $\deg \overline{F} \le \deg F < |D/P|$. $\square$

3 Construction of irreducible polynomials with prescribed valuations

In this section, we construct irreducible polynomials whose minimal valuations on a given residue class can be prescribed. We will use those in the proof of the main result to build an integer-valued polynomial that has factorization lengths of our choice.

The following remark seems to be a well-known result, but we could not find a quotable reference, so we included a proof.

Remark 3.1

Let $(K,\mathsf v)$ be a discretely valued field and let F be a polynomial of degree n over K. If the Newton-polygon of F has just one slope $\frac{\lambda }{n}$ such that $\gcd (\lambda ,n)=1$, then F is irreducible over K.

Proof

Let F be as in the statement and assume to the contrary that F is reducible. Let L be a splitting field of F and let $\mathsf w$ be an extension of $\mathsf v$ to L. Now let $\alpha \in L$ be a root of F, then $[K(\alpha ):K]=d<n$ and by [11, Ch. II, Proposition 6.3], $\mathsf w(\alpha )=-\frac{\lambda }{n}$. On the other hand, $\mathsf w(\alpha )\in \frac{1}{d}\mathbb {Z}$, so there is $m\in \mathbb {Z}$ with $m=\frac{\lambda d}{n}$, contradicting $\gcd (\lambda , n)=1$. $\square$

The following fact was proven by Samuel [14, Proposition 13] for Noetherian domains and Halter–Koch gave an alternative proof for Dedekind domains, see [9, Proposition 2.1]. The proof for Krull domains is identical to the latter but for convenience of the reader, we include it here.

Remark 3.2

Let D be Krull domain and $\Sigma$ a positive integer. Then there are only finitely many height-one prime ideals Q of D with $|D/Q| \le \Sigma$.

Proof

Assume to the contrary that there exist infinitely many height-one prime ideals of D with $|D/Q| = \Sigma$, and let $a \in D \setminus \{0\}$. Then, by definition of a Krull domain, there exist infinitely many height-one primes Q of D such that $|D/Q| = \Sigma$ and $a \notin Q$. For each such Q, we obtain $a^{q-1} - 1 \in Q$ because the multiplicative group of the field D/Q has exactly $q-1$ elements. Again by definition of a Krull domain, $a^{q-1} - 1= 0$, whence $a^{q-1} = 1$. So, every non-zero element of D is a $(q-1)$-st root of unity, which is a contradiction. $\square$

Lemma 3.3

Let D be a Krull domain admitting a prime element $\pi$ such that $D/\pi D$ is finite. Denote by K the quotient field of D and let $\Sigma$ be a positive integer. Let $\mathsf v_Q: K^\times \rightarrow \mathbb {Z}$ be the normalized valuation on K with respect to $Q \in \mathfrak {X}(D)$ and denote $\textsf{v} = \textsf{v}_{\pi D}$. Let $\pi D = R_1,\ldots ,R_q$ be the residue classes with respect to $\pi D$. For each $k \in \{1,\ldots ,q\}$, choose $r_k \in R_k$ such that $r_1=\pi$ and $r_s - r_1 \in Q$ for all $s\in \{1,\ldots ,q\}$ and all $Q \in \mathfrak {X}(D) \setminus \{\pi D\}$ with $|D/Q| \le \Sigma$ (this is possible by Remark 3.2 and the approximation theorem for Krull domains).

Then, for every $s\in \{1,\ldots ,q\}$ and every positive integer $n \ge 2$, there exist infinitely many polynomials $F \in D[X]$ of degree n that are pairwise non-associated and irreducible over K with the following properties:

• (1) $\min \{ \mathsf v(F(a)) \mid a\in R_s\}=\mathsf v(F(r_s))=n$.

• (2) For all $a\in D\setminus R_s$, we have $\mathsf v(F(a))=0$.

• (3) $\textsf{v}_Q(F(0)) = 0$ for all $Q \in \mathfrak {X}(D) \setminus \{ \pi D\}$ with $|D/Q| \le \Sigma$.

Proof

First, let $s=1$, so $R_s=\pi D$. Setting $c = \pi ^{n+1}$ and $m \ge n+1$, we claim that $F=X^n + \pi ^m X^{n-1} +\sum _{i=0}^{n-2} c X^i$ has the desired properties. Let $a\in \pi D$. Then

$$\begin{aligned} \mathsf v(F(a))&\ge \min \{\mathsf v(c),\mathsf v(c)+\mathsf v(a),\ldots , m +(n-1)\mathsf v(a), n\mathsf v(a)\}\\&\ge n=\min \{\mathsf v(c),\mathsf v(c)+\mathsf v(r_1),\ldots , m+(n-1)\mathsf v(r_1), n\mathsf v(r_1)\}=\mathsf v(F(r_1)), \end{aligned}$$

where the last equality follows from the fact that the minimum is only accepted at the value $n\mathsf v(r_1)$, which proves (1).

For (2), let $a\in D\setminus \pi D$. Then

$$\begin{aligned} \mathsf v(F(a))\ge \min \{\mathsf v(c), \mathsf v(c)+\mathsf v(a),\ldots , m+(n-1)\mathsf v(a), n\mathsf v(a)\}=0. \end{aligned}$$

Since this minimum is only accepted at the value $n\mathsf v(a)$, we conclude $\mathsf v(F(a))=0$.

(3) is trivial as $F(0) = \pi ^{n+1}$. To see that F is irreducible, note that its Newton-polygon has exactly one slope, which is $-\frac{n+1}{n}$, and use Remark 3.1 for the discrete valuation ring $D_{\pi D}$.

The fact that we can vary m over all integers $\ge n+1$ gives infinitely many such F pairwise non-associated over K.

Now let $s\ne 1$. We set $F'(X)=F(X-r_s+r_1)$. Since $X\mapsto X-r_s+r_1$ induces an isomorphism $K[X]\rightarrow K[X]$, we conclude that all these $F'$ are irreducible and pairwise non-associated over K. Moreover,

$$\begin{aligned} \mathsf v(F'(r_s))=\mathsf v(F(r_1))=n \end{aligned}$$

and if $a\in R_s$, then $a-r_s\in \pi D$, so $(a-r_s)+r_1\in \pi D$, whence

$$\begin{aligned} \mathsf v(F'(a))=\mathsf v(F(a-r_s+r_1))\ge \mathsf v(F(r_1))=n=\mathsf v(F'(r_s)). \end{aligned}$$

On the other hand, if $a\in D\setminus R_s$, then $a-r_s\notin \pi D$, thus $(a-r_s)+r_1\notin \pi D$ and it follows

$$\begin{aligned} \mathsf v(F'(a))=\mathsf v(F(a-r_s+r_1))=0. \end{aligned}$$

Finally, $\mathsf v_Q(F'(0)) =\mathsf v_Q(F(r_1-r_s)) =\mathsf v_Q(c) = 0$ for $Q \ne \pi D$ with $|D/Q| \le \Sigma$. $\square$

4 Combinatorial toolbox

The following combinatorial section is identical with the one in [6], but to make the manuscript readable without consulting [6] all the time, we include the notational conventions and the final result without proof here.

Notation 4.1

Let n be a positive integer. We write $[n] = \{1,\ldots ,n\}$.

Notation 4.2

Let $2 \le k$, $2\le n_1 \le \cdots \le n_k$ be integers, $i,j \in \{1, \ldots ,k\}$ with $i < j$, $S \subseteq [n_i]$, and $T \subseteq [n_j]$. We set

$$\begin{aligned} H_{i,j}(S,T){} & {} = H_{j,i}(T,S) = [n_1] \times \cdots \times [n_{i-1}] \times S \times [n_{i+1}]\\{} & {} \quad \ \times \cdots \times [n_{j-1}] \times T \times [n_{j+1}] \times \cdots \times [n_k].\end{aligned}$$

For $s \in [n_i]$, we define $H_{i,j}(s,T)=H_{i,j}(\{s\},T)$. Moreover, we write $H_{i,j}(S,[n_j]) = H_i(S)$, $H_i(s) = H_i(\{s\})$, and $H_{i,j}(s,t) = H_{i,j}(\{s\},\{t\})$.

Note that the $H_i(s)$ are $(k-1)$-dimensional hyperplanes in the grid $[n_1] \times \cdots \times [n_k]$. Analogously, the $H_{i,j}(s,t)$ are $(k-2)$-dimensional hyperplanes.

Notation 4.3

Let $k >1$ and $1< n_1 \le \cdots \le n_k$ be integers. By $\mathbb {Q}^{n_1\times \cdots \times n_k}$ we denote the set of all $(n_1 \times \cdots \times n_k)$–arrays, that is, the k-dimensional analogues of matrices over $\mathbb {Q}$. Let $M \in \mathbb {Q}^{n_1\times \cdots \times n_k}$. For $i \in [n_1] \times \cdots \times [n_k]$, we write $M_i$ for the entry of M indexed by i.

For $I \subseteq [n_1] \times \cdots \times [n_k]$, let $Z_I = \{M \in \mathbb {Q}^{n_1\times \cdots \times n_k} \mid \sum _{i \in I} M_i = 0\}$. Moreover,

$$\begin{aligned} Z:= \bigcap _{\begin{array}{c} \ell \in \{1,\ldots , k\} \\ r \in [n_\ell ] \end{array}} Z_{H_\ell (r)}.\end{aligned}$$

For instance, if $k = 2$, then Z is the set of all $(n_1 \times n_2)$–matrices with all row and column sums equal to 0.

Notation 4.4

Let $k,q >1$ and $1< n_1 \le \cdots \le n_k$ be integers. For $I_1,\ldots , I_q \subseteq [n_1] \times \cdots \times [n_k]$, let

$$\begin{aligned} Z_{I_1,\ldots ,I_q} = \{(M_1,\ldots ,M_q) \in (\mathbb {Q}^{n_1\times \cdots \times n_k})^q \mid \forall i,j \ \sum _{m \in I_i} (M_i)_m = \sum _{m\in I_j} (M_j)_m\}. \end{aligned}$$

Moreover, we define

$$\begin{aligned} \overline{Z}:= \bigcap _{\begin{array}{c} \ell \in \{1,\ldots , k\} \\ r \in [n_\ell ] \end{array}} Z_{H_\ell (r),\ldots , H_\ell (r)}.\end{aligned}$$

For $s\in \mathbb {Q}$, we denote by (s) the array all of whose entries are s.

Note that [6, Proposition 4.10] is the special case of Proposition 4.5 below in case $q = 2$.

Proposition 4.5

Let $k,q>1$ and $1 < n_1 \le \cdots \le n_k$ be integers. Then there exists $(M_1,\ldots ,M_q) \in \overline{Z}$ such that $(M_1,\ldots ,M_q) \in Z_{I_1,\ldots ,I_q}$ only if $I_1=\cdots =I_q$ is a disjoint union of hyperplanes, that is, $I_1 = H_\ell (S)$ for some $\ell \in \{1,\ldots ,k\}$ and $S \subseteq [n_\ell ]$. Moreover, the $M_i$ can be chosen such that all of their entries are positive integers.

5 Sets of lengths of integer-valued polynomials over a DVR

Lemma 5.1

Let D be a Krull domain with quotient field K and let I be a finite set. Suppose that D admits a prime element $\pi$ and denote by $\textsf{v}$ its normalized valuation on K. Let $f\in {{\,\textrm{Int}\,}}(D)$ be of the form

$$\begin{aligned} f=\frac{\prod _{i\in I}f_i}{\pi ^e}, \end{aligned}$$

where, for every $i\in I$, $f_i\in D[X]$ is irreducible over K, $e\in \mathbb {N}$, and such that

$$\begin{aligned} \min \{\mathsf v_Q(f(a))\mid a\in D\}=0 \text { for all } Q\in \mathfrak {X}(D). \end{aligned}$$

If $f=g_1\cdots g_m$ is a decomposition into non-units in ${{\,\textrm{Int}\,}}(D)$, then every $g_j$ is, up to multiplication by units of D, of the form

$$\begin{aligned} g_j=\frac{\prod _{i\in I_j}f_i}{\pi ^{e_j}}, \end{aligned}$$

where $I_1,\ldots ,I_m$ are non-empty and form a partition of I and $\sum _{j=1}^m e_j=e$.

Proof

Let $f=g_1\cdots g_m$, then clearly no $g_j$ is a constant, because of the condition on the minima of valuations of evaluations of f. From the irreducibility of the $f_i$, it follows that every $g_j=\frac{\prod _{i\in I_j}f_i}{a_j}$ with $a_j\in K$, where the $I_j$ are non-empty and form a partition of I and $\prod _{j=1}^m a_j=\pi ^e$. Now it is clear that $a_j=\pi ^{e_j}$, where $\sum _{j=1}^m e_j=e$. $\square$

Lemma 5.2

Let D be a Krull domain admitting a prime element $\pi$ such that $D/\pi D$ is finite. Let K be the quotient field of D and $\mathsf v: K^\times \rightarrow \mathbb {Z}$ the normalized valuation with respect to $\pi$. By $R_1,\ldots ,R_q$ denote the residue classes of D modulo $\pi D$. Let k be a positive integer and $1 < n_1 \le \cdots \le n_k$ integers.

Let $\Sigma$ be a positive integer and let $r_1,\ldots ,r_q$ be a complete system of residues with $r_i \in R_i$ such that $r_1=\pi$ and $r_s - r_1 \in Q$ for all $s\in \{1,\ldots ,q\}$ and all $Q \in \mathfrak {X}(D) \setminus \{\pi D\}$ with $|D/Q| \le \Sigma$ (this is possible by Remark 3.2 and the approximation theorem for Krull domains).

For $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$, let $F_i^{(s)} \in D[X]$ irreducible over K such that

• (i) $\min \{\textsf{v}(F_i^{(s)}(a)) \mid a \in R_s\} = \textsf{v}(F_i^{(s)}(r_s))$ for all $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$,

• (ii) $\min \{\textsf{v}(F_i^{(s)}(a)) \mid a \in R_t\} = 0 = \textsf{v}(F_i^{(s)}(r_t))$ for all $i \in [n_1] \times \cdots \times [n_k]$, $s \in \{1, \ldots , q\}$, and $t \in \{1, \ldots , q\} \setminus \{s\}$,

• (iii) $\sum _{i \in [n_1] \times \dots \times [n_k]} \textsf{v}(F_i^{(s)}(r_s)) = \sum _{i \in [n_1] \times \dots \times [n_k]} \textsf{v}(F_i^{(t)}(r_t))$ for all $s,t \in \{1,\ldots ,q\}$,

• (iv) for each $Q \in \mathfrak {X}(D) \setminus \{\pi D\}$, there exists $a \in D$ such that $\textsf{v}_Q(F_i^{(s)}(a)) = 0$ for all $i \in [n_1] \times \cdots \times [n_k]$, and $s \in \{1, \ldots , q\}$, and

• (v) $\sum _{i,s} \deg F_i^{(s)} \le \Sigma$.

Define $e = \sum _{i \in [n_1] \times \dots \times [n_k]} \textsf{v}(F_i^{(s)}(r_s))$ for some $s \in \{1,\ldots ,q\}$ (this is independent from s by (iii)) and set

$$\begin{aligned} H = \frac{\prod _{s = 1}^q \prod _{i \in [n_1] \times \cdots \times [n_k]} F_i^{(s)}}{\pi ^e}. \end{aligned}$$

Then $H \in {{\,\textrm{Int}\,}}(D)$. Furthermore, H is a product of two non-units in ${{\,\textrm{Int}\,}}(D)$ if and only if there exist non-empty $I_1,\ldots ,I_q \subsetneqq [n_1] \times \cdots \times [n_k]$ such that $\sum _{i \in I_s} \textsf{v}(F_i^{(s)}(r_s)) = \sum _{i \in I_t} \textsf{v}(F_i^{(t)}(r_t))$ for all $s,t \in \{1,\ldots ,q\}$.

Proof

Clearly H is integer-valued over D. If there exist $I_1,\ldots ,I_q$ as in the lemma, define $e' = \sum _{i \in I_s} \textsf{v}(F_i^{(s)}(r_s))$ for some $s \in \{1,\ldots ,q\}$. Then clearly

$$\begin{aligned} H = \frac{\prod _{s = 1}^q \prod _{i \in I_s} F_i^{(s)}}{\pi ^{e'}} \cdot \frac{\prod _{s = 1}^q \prod _{i \in ([n_1] \times \cdots \times [n_k]) \setminus I_s} F_i^{(s)}}{\pi ^{e-e'}} \end{aligned}$$

is a decomposition into two non-units in ${{\,\textrm{Int}\,}}(D)$.

Conversely, if $H = H_1 \cdot H_2$ is a decomposition of H where $H_1$ and $H_2$ are non-units of ${{\,\textrm{Int}\,}}(D)$, then, by Lemma 5.1, there exist non-empty $I_1,\ldots ,I_q \subsetneqq [n_1]\times \cdots \times [n_k]$ such that

$$\begin{aligned} H_1 = \frac{\prod _{s = 1}^q \prod _{i \in I_s} F_i^{(s)}}{\pi ^{e'}} \quad \text { and }\quad H_2 = \frac{\prod _{s = 1}^q \prod _{i \in ([n_1] \times \cdots \times [n_k]) \setminus I_s} F_i^{(s)}}{\pi ^{e-e'}} \end{aligned}$$

for some $e' \in \{0,\ldots ,e\}$ (note that the assumptions of Lemma 5.1 are satisfied by (i), (ii), (iii), and (iv) together with the definition of e).

To see that the $I_1,\ldots ,I_q$ are indeed non-empty proper subsets, assume, for instance, $I_1 = \emptyset$. If $e' >0$, then $H_1$ cannot be integer-valued as $\textsf{v}(H_1(r_1)) = -e'$ by (ii). Now suppose $e' = 0$. As $H_1$ is not a unit, there exists some s such that $I_s \ne \emptyset$. Therefore, by (i), (ii), and (iii), $\textsf{v}(H_2(r_s)) < e = e-e' = \textsf{v}(\pi ^{e-e'})$. So, in this case, $H_2$ cannot be integer-valued. By exchanging the roles of $H_1$ and $H_2$, one sees in the same way that $I_1$ is also a proper subset.

Now, assume to the contrary that $\sum _{i \in I_s} \textsf{v}(F_i^{(s)}(r_s)) \ne \sum _{i \in I_t} \textsf{v}(F_i^{(t)}(r_t))$ for some $s,t \in \{1,\ldots ,q\}$, say $\sum _{i \in I_s} \textsf{v}(F_i^{(s)}(r_s)) > \sum _{i \in I_t} \textsf{v}(F_i^{(t)}(r_t))$. Since $H_1$ is an integer-valued polynomial on D, it follows that

$$\begin{aligned} \sum _{i \in I_s} \mathsf v(F_i^{(s)}(r_s)) > \sum _{i \in I_t} \mathsf v(F_i^{(s)}(r_t)) \ge e'.\end{aligned}$$

Hence we get

$$\begin{aligned} \sum _{i \in ([n_1]\times \cdots \times [n_k]) \setminus I_s} \mathsf v(F_i^{(s)}(r_s)) < \sum _{i \in ([n_1]\times \cdots \times [n_k]) \setminus I_t} \mathsf v(F_i^{(s)}(r_t)) \le e - e', \end{aligned}$$

which is a contradiction because $H_2 \in {{\,\textrm{Int}\,}}(D)$. $\square$

By a q-tuple of ordered partitions of sets $I_1,\ldots ,I_q$, we mean an equivalence class of tuples $((I_1^{(1)},\ldots ,I_1^{(\ell )}),\ldots ,(I_q^{(1)},\ldots ,I_q^{(\ell )}))$, where the $I_s^{(\lambda )}$ form a partition of $I_s$, under the equivalence relation where

$$\begin{aligned} ((I_1^{(1)},\ldots ,I_1^{(\ell )}),\ldots ,(I_q^{(1)},\ldots ,I_q^{(\ell )})) \end{aligned}$$

is identified with

$$\begin{aligned} ((I_1^{(\sigma (1))},\ldots ,I_1^{(\sigma (\ell ))}),\ldots ,(I_q^{(\sigma (1))},\ldots ,I_q^{(\sigma (\ell ))})) \end{aligned}$$

for all permutations $\sigma$ of $\{1,\ldots ,\ell \}$. The positive integer $\ell$ is called the length of the q-tuple of ordered partitions.

Lemma 5.3

Let D be a Krull domain admitting a prime element $\pi$ such that $D/\pi D$ is finite. Let K be the quotient field of D and $\mathsf v: K^\times \rightarrow \mathbb {Z}$ the normalized valuation with respect to $\pi$. By $R_1,\ldots ,R_q$ denote the residue classes of D modulo $\pi D$. Let k be a positive integer and $1 < n_1 \le \cdots \le n_k$ integers.

Let $\Sigma$ be a positive integer and let $r_1,\ldots ,r_q$ be a complete system of residues with $r_i \in R_i$ such that $r_1=\pi$ and $r_s - r_1 \in Q$ for all $s\in \{1,\ldots ,q\}$ and all $Q \in \mathfrak {X}(D) \setminus \{\pi D\}$ with $|D/Q| \le \Sigma$ (this is possible by Remark 3.2 and the approximation theorem for Krull domains).

For $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$, let $F_i^{(s)} \in D[X]$ irreducible over K such that

• (i) $\min \{\textsf{v}(F_i^{(s)}(a)) \mid a \in R_s\} = \textsf{v}(F_i^{(s)}(r_s))$ for all $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$,

• (ii) $\min \{\textsf{v}(F_i^{(s)}(a)) \mid a \in R_t\} = 0 = \textsf{v}(F_i^{(s)}(r_t))$ for all $i \in [n_1] \times \cdots \times [n_k]$, $s \in \{1, \ldots , q\}$, and $t \in \{1, \ldots , q\} \setminus \{s\}$,

• (iii) $\sum _{i \in [n_1] \times \dots \times [n_k]} \textsf{v}(F_i^{(s)}(r_s)) = \sum _{i \in [n_1] \times \dots \times [n_k]} \textsf{v}(F_i^{(t)}(r_t))$ for all $s,t \in \{1,\ldots ,q\}$,

• (iv) for each $Q \in \mathfrak {X}(D) \setminus \{\pi D\}$, there exists $a \in D$ such that $\textsf{v}_Q(F_i^{(s)}(a)) = 0$ for all $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$, and

• (v) $\sum _{i,s} \deg F_i^{(s)} \le \Sigma$.

Define $e = \sum _{i \in [n_1] \times \dots \times [n_k]} \textsf{v}(F_i^{(s)}(r_s))$ for some $s \in \{1,\ldots ,q\}$ (this is independent from s by (iii)) and set

Then, for every $\ell >0$, a bijective correspondence between, on the one hand, q-tuples of ordered partitions

$$\begin{aligned} ((I_1^{(1)},\ldots ,I_1^{(\ell )}),\ldots ,(I_q^{(1)},\ldots ,I_q^{(\ell )})), \end{aligned}$$

where each $(I_s^{(1)},\ldots ,I_s^{(\ell )})$ is a partition of $[n_1]\times \cdots \times [n_k]$, satisfying

• (1) $e_j:= \sum _{i \in I_s^{(j)}} \textsf{v}(F_i^{(s)}(r_s)) = \sum _{i \in I_t^{(j)}} \textsf{v}(F_i^{(t)}(r_t))$ for all $s,t \in \{1,\ldots ,q\}$ and $j \in \{1, \ldots , \ell \}$,

• (2) for all $j \in \{1, \ldots , \ell \}$ and non-empty $J_1^{(j)} \subsetneqq I_1^{(j)},\ldots , J_q^{(j)} \subsetneqq I_q^{(j)}$, there exist $s,t \in \{1,\ldots ,q\}$ such that $\sum _{i \in J_s^{(j)}} \textsf{v}(F_i^{(s)}(r_s)) \ne \sum _{i \in J_t^{(j)}} \textsf{v}(F_i^{(t)}(r_t))$,

and, on the other hand, essentially different factorizations of H into $\ell$ irreducible elements of ${{\,\textrm{Int}\,}}(D)$ is given by

$$\begin{aligned}&((I_1^{(1)},\ldots ,I_1^{(\ell )}),\ldots ,(I_q^{(1)},\ldots ,I_q^{(\ell )})) \mapsto \frac{\prod _{s=1}^q \prod _{i \in I_s^{(1)}} F_i^{(s)}}{\pi ^{e_1}} \cdots \frac{\prod _{s=1}^q \prod _{i \in I_s^{(\ell )}} F_i^{(s)}}{\pi ^{e_\ell }}. \end{aligned}$$

Proof

This follows immediately from Lemma 5.2. $\square$

Theorem 1

Let D be a Krull domain admitting a prime element $\pi$ such that $D/\pi D$ is finite.

Then ${{\,\textrm{Int}\,}}(D)$ has full system of multisets of lengths, i.e., for all positive integers k and all integers $1 < n_1 \le \cdots \le n_k$, there exists an integer-valued polynomial $H \in {{\,\textrm{Int}\,}}(D)$ which has precisely k essentially different factorizations into irreducible elements of ${{\,\textrm{Int}\,}}(D)$ whose lengths are exactly $n_1,\ldots ,n_k$.

Proof

Let K be the quotient field of D, $\mathsf v: K^\times \rightarrow \mathbb {Z}$ the normalized valuation of the prime element $\pi \in D$. By $\pi D = R_1,\ldots ,R_q$ denote the residue classes of D modulo $\pi D$.

By Proposition 4.5, there exists a q-tuple of arrays $(M_1,\ldots ,M_q) \in (\mathbb {Q}^{n_1\times \cdots \times n_k})^q$ where all arrays have positive integers as entries and such that $(M_1,\ldots ,M_q) \in \overline{Z}$ with $(M_1,\ldots ,M_q) \in Z_{I_1,\ldots ,I_q}$ only if $I_1=\cdots =I_q$ is a disjoint union of hyperplanes, that is, $I_1 = H_\ell (S)$ for some $\ell \in \{1,\ldots ,k\}$ and $S \subseteq [n_\ell ]$ (see Notation 4.2, 4.3, and 4.4). Recall that $(M_s)_i$ for $i \in [n_1]\times \cdots \times [n_k]$, denotes the i-th entry of $M_s$.

We set $\Sigma =\sum _{i\in [n_1] \times \cdots \times [n_k]} (M_s)_i$ which is independent of the choice of s. Let $r_1,\ldots ,r_q$ be a complete system of residues with $r_i \in R_i$ such that $r_1=\pi$ and $r_s - r_1 \in Q$ for all $s\in \{1,\ldots ,q\}$ and all $Q \in \mathfrak {X}(D) \setminus \{\pi D\}$ with $|D/Q| \le \Sigma$ (this is possible by Remark 3.2 and the approximation theorem for Krull domains).

If $k=1$, then $H= X^{n_1}$ has the desired property. So let $k\ge 2$. By Lemma 5.3, it suffices to construct polynomials $F_i^{(s)}$ as in the hypothesis of this lemma such that there are exactly k different q-tuples of partitions of $[n_1] \times \cdots \times [n_k]$ and the cardinalities of the partitions are exactly $n_1,\ldots ,n_k$. Then H as in the equation ($*$) has the desired property.

By Lemma 3.3, there exist, for $i \in [n_1]\times \ldots \times [n_k]$ and $s \in \{1,\ldots ,q\}$, polynomials $F_i^{(s)}\in D[X]$ which are all pairwise non-associated and irreducible over K with the following properties:

• (i) $\min \{\textsf{v}(F_i^{(s)}(a)) \mid a \in R_s\} = \textsf{v}(F_i^{(s)}(r_s)) = (M_s)_i=\deg (F_i^{(s)})$ for all $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$,

• (ii) $\min \{\textsf{v}(F_i^{(s)}(a)) \mid a \in R_t\} = 0 = \textsf{v}(F_i^{(s)}(r_t))$ for all $i \in [n_1] \times \cdots \times [n_k]$, $s \in \{1, \ldots , q\}$, and $t \in \{1, \ldots , q\} \setminus \{s\}$, and

• (iii) $\textsf{v}_Q(F_i^{(s)}(0)) = 0$ for all $Q \in \mathfrak {X}(D) \setminus \{ \pi D\}$ with $|D/Q| \le \Sigma$ and all $i \in [n_1] \times \cdots \times [n_k]$ and $s \in \{1, \ldots , q\}$.

By the choice of the $M_s$, the only admissible (in the sense of Lemma 5.3, that is, satisfying (1) and (2) of this lemma) q-tuples of ordered partitions of $\ell$ copies of the index set $[n_1]\times \cdots \times [n_k]$, namely those that correspond to factorizations into irreducibles, are the ones of the form $((H_r(1),\ldots , H_r(n_r)), \ldots , (H_r(1), \ldots , H_r(n_r)))$ for $r \in \{1,\ldots ,k\}$. These are exactly k many of cardinalities $n_1,\ldots ,n_k$. $\square$

Corollary 5.4

Let D be a unique factorization domain. Exactly one of the following holds:

• (1) ${{\,\textrm{Int}\,}}(D) = D[X]$ is a unique factorization domain.

• (2) ${{\,\textrm{Int}\,}}(D)$ has full system of multisets of lengths, i.e., for all positive integers k and all integers $1 < n_1 \le \cdots \le n_k$, there exists an integer-valued polynomial $H \in {{\,\textrm{Int}\,}}(D)$ which has precisely k essentially different factorizations into irreducible elements of ${{\,\textrm{Int}\,}}(D)$ whose lengths are exactly $n_1,\ldots ,n_k$.

Moreover, (1) holds if and only if the residue field of each height-one prime ideal in D is infinite.

Proof

By [3, Corollary I.3.15], ${{\,\textrm{Int}\,}}(D) = D[X]$ if and only if the residue field of each height-one prime ideal in D is infinite, in which case ${{\,\textrm{Int}\,}}(D)$ is a UFD. In the other case, use Theorem 1. $\square$