Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integer-valued-polynomials]

For questions related to integer valued polynomials and their properties such as fixed divisor of a polynomial or representation in binomial base.

Integer valued polynomial is any polynomial $f(x)$ whose values $f(n)$ are integers for all integers $n$. Integer valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.

All integer valued polynomials can be written uniquely in form $$ f(x)=a_{n}\binom {x}{n}+a_{n-1}\binom {x}{n-1}+\dots+a_0\binom {x}{0}$$

Source: Integer-valued polynomial.

26 questions
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How does one prove that the ring of integer-valued polynomials $\text{Int}(\mathbb{Z})$ is not Noetherian?

How does one prove that the ring of integer-valued polynomials $\text{Int}(\mathbb{Z})$ is not Noetherian? I let $(1, f_1, ..., f_n,...)$ be the $\mathbb{Z}$-basis of $\text{Int}(\mathbb{Z})$, the ring of rational polynomials sending $\mathbb{Z}$…
byesac
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Ring of integer-valued polynomials Int(X) is not isomorphic to Z[X]

Write $\operatorname{Int}(\mathbb{Z})$ for the set of polynomials $p$ mapping integers to integers, i.e., $p(\mathbb{Z}) \subseteq \mathbb{Z}$. I have shown that $\operatorname{Int}(\mathbb{Z})$ is a subring of $\mathbb{Q}[X]$. I am now asked to…
Jingjie Yang
  • 462
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2 answers

Integer valued polynomial through some known points

I have two questions, but I'll put both of them here since they are closely related: An integer valued polynomial $P(x)$ is a polynomial whose value $P(n)\in\mathbb{N}$ for every $n\in\mathbb{N}$. 1. I'm given a set of points with integer…
Alessandro Codenotti
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Valuations of integer valued polynomials

Consider the ring $R=\text{Int}(\mathbb Z):=\{p(x)\in \mathbb Q[x]\ |\ p(n)\in \mathbb Z, \forall n\in \mathbb Z \}$. Let $K$ denote the fraction field of $R$. Fix an $a\in \mathbb Z$ and let $P$ be the prime ideal defined by $P:=\{q(x)\in R\ | \…
user386633
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is the hilbert polynomial integer-valued everywhere?

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, finitely generated over $R_0$ with $R_0$ local Artinian. Let $M$ be a finite $R$-module of Krull dimension $d$. It is known that the Hilbert function $H(M,n) = \operatorname{length} (M_n)$ coincides…
Manos
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Integer-valued polynomials divisible by 2 or 3

I would like some help with the following problem: Suppose that $p \in \operatorname{Int}(\mathbb{Z}) = \{p \in \mathbb{Q}[X] : p(\mathbb{Z}) \subseteq \mathbb{Z}\}$ is such that for all $z \in \mathbb{Z}$, either $2$ divides $p(z)$ or $3$ divides…
Jingjie Yang
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2 answers

Is the Lagrange polynomial integer-valued for points with consecutive integer x-values?

What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to extend every finite integer sequence? Let $S$ be a…
ajm475du
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Integer valued polynomials in two variables

The ring of integer valued polynomials, $\{ f \in \mathbb{Q}[x] : f(\mathbb{Z}) \subseteq \mathbb{Z} \}$ is fairly well-known to be generated as Abelian group by the binomial coefficients, $f_k(n) = \binom{n}{k}$ of degree $k$. What about the…
Jack Schmidt
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A class of integer valued polynomials

Computations suggest that the polynomials $$p_n(x)=\prod_{k=1}^n \left(\frac{x+k}{k}\right)^{\min(k,n-k)}$$ are integer valued. Is there a simple proof of this fact? Edit For a positive integer $m$ this reduces to…
Johann Cigler
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Avoiding brute force: determining when a specific polynomial in $\mathbb{Q}[x]$ is an integer for any integer $x$

I have to prove that $\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n$ is an integer for any $n$. I solved this by brute-force, exhausting all the possibilities methods. I was wondering if there was a way to solve this at-a-glance with some sort of…
no lemon no melon
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Given an integer-valued polynomial, can its sequential values be used to determine whether it is reducible?

Let $p(x)$ be an integer-valued polynomial of degree $n$. Is it possible to use up to $n$ consecutive values to potentially identify $p(x)$ as irreducible over the set of integer-valued polynomials? I managed to prove that for $n=2$ there is a…
abiessu
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Questions on integer-valued polynomials

An integer-valued polynomial or numerical polynomial is a polynomial $f \in \mathbb Q[x]$ with the property that $f(\mathbb Z)\subseteq \mathbb Z$. The set of numerical polynomials forms a subring $\mathcal N$ of $\mathbb Q[x]$. Obviously $\mathbb…
Bruno Joyal
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Integer-valued polynomial

Let $f(x) \in \mathbb{Q}[x]$, and suppose $f(n)$ is an integer for all large integer $n$. Prove that $f(n)$ is an integer for small positive integers $n$. I read the answer from here is the hilbert polynomial integer-valued everywhere?, but I'm…
dh16
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A monic integer-valued polynomial in $\mathbb{Q}[x]$ must be in $\mathbb{Z}[x]$

Is it true that if $f(x) \in \mathbb{Q}[x]$ is a $\textbf{monic}$ polynomial such that $f(k) \in \mathbb{Z}$ for all (sufficiently large) $k\in \mathbb{Z}$ then $f(x) \in \mathbb{Z}[x]$? I am aware of non-monic counterexamples such as $\binom{x}{n}$…
DesmondMiles
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Show that $P_k(X) = \frac{1}{k!} X (X - 1) ... (X - k + 1)$ is an integer-valued polynomial.

For $k \in \mathbb{N}$, let $P_k(X) = \frac{1}{k!} X (X - 1)(X - 2) ... (X - k + 1)$. Show that these are integer-valued polynomials, i.e. $P_k(X) \in \left\{f \in \mathbb{Q}[X] \mid f(\mathbb{Z}) \subset \mathbb{Z}\right\}$ for all $k \in…
Kasper Cools
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