ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Integer-valued polynomial</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Integer-valued polynomial</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>integer-valued polynomial</b> (also known as a <b>numerical polynomial</b>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(t)}">

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<annotation encoding="application/x-tex">{\displaystyle P(t)}</annotation>

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</math></span><img src="./941c1fb8073a146dc9938ef2aa43c2a84f6b2c97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.394ex; height:2.843ex;" alt="{\displaystyle P(t)}" loading="lazy"></span> is a <a href="Polynomial" title="Polynomial">polynomial</a> whose value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}">

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<annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation>

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</math></span><img src="./5e303d2c14cd399b6f52b468c9fd44a542bed422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}" loading="lazy"></span> is an <a href="Integer" title="Integer">integer</a> for every integer <i>n</i>. Every polynomial with integer <a href="Coefficient" title="Coefficient">coefficients</a> is integer-valued, but the converse is not true. For example, the polynomial

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<annotation encoding="application/x-tex">{\displaystyle P(t)={\frac {1}{2}}t^{2}+{\frac {1}{2}}t={\frac {1}{2}}t(t+1)}</annotation>

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</math></span><img src="./efc5a4241bb705f177031d3b6875edd5d54815d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.652ex; height:5.176ex;" alt="{\displaystyle P(t)={\frac {1}{2}}t^{2}+{\frac {1}{2}}t={\frac {1}{2}}t(t+1)}" loading="lazy"></span></dd></dl>

<p>takes on integer values whenever <i>t</i> is an integer. That is because one of <i>t</i> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t+1}">

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<annotation encoding="application/x-tex">{\displaystyle t+1}</annotation>

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</math></span><img src="./ab2785d8415d6902b0c93efe1419c4bc3ce4643d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.842ex; height:2.343ex;" alt="{\displaystyle t+1}" loading="lazy"></span> must be an <a href="Even_number" class="mw-redirect" title="Even number">even number</a>. (The values this polynomial takes are the <a href="Triangular_number" title="Triangular number">triangular numbers</a>.)

</p><p>Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2></div>

<p>The class of integer-valued polynomials was described fully by <a href="George_P%C3%B3lya" title="George Pólya">George Pólya</a> (<a href="#CITEREFPólya1915">1915</a>). Inside the <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} [t]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} [t]}</annotation>

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</math></span><img src="./209b3d34c2a122140b66a289f6bc4fd2340d5d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.941ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} [t]}" loading="lazy"></span> of polynomials with <a href="Rational_number" title="Rational number">rational number</a> coefficients, the <a href="Subring" title="Subring">subring</a> of integer-valued polynomials is a <a href="Free_abelian_group" title="Free abelian group">free abelian group</a>. It has as <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> the polynomials

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<annotation encoding="application/x-tex">{\displaystyle P_{k}(t)=t(t-1)\cdots (t-k+1)/k!}</annotation>

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</math></span><img src="./445f62f323f79d6e525d0a07fab06b3505464e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.041ex; height:2.843ex;" alt="{\displaystyle P_{k}(t)=t(t-1)\cdots (t-k+1)/k!}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle k=0,1,2,\dots }</annotation>

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</math></span><img src="./40e06b69a163e607d10bcd0874c092df7b3a5de2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.622ex; height:2.509ex;" alt="{\displaystyle k=0,1,2,\dots }" loading="lazy"></span>, i.e., the <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>. In other words, every integer-valued polynomial can be written as an integer <a href="Linear_combination" title="Linear combination">linear combination</a> of binomial coefficients in exactly one way. The proof is by the method of <a href="Difference_operator" class="mw-redirect" title="Difference operator">discrete Taylor series</a>: binomial coefficients are integer-valued polynomials, and conversely, the discrete difference of an integer series is an integer series, so the discrete <a href="Taylor_series" title="Taylor series">Taylor series</a> of an integer series generated by a polynomial has integer coefficients (and is a finite series).

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<div class="mw-heading mw-heading2"><h2 id="Fixed_prime_divisors">Fixed prime divisors</h2></div>

<p>Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials <i>P</i> with integer coefficients that always take on even number values are just those such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P/2}">

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<annotation encoding="application/x-tex">{\displaystyle P/2}</annotation>

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</math></span><img src="./9c310e8d598049ebbb4045c4701941b43bfdec49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.07ex; height:2.843ex;" alt="{\displaystyle P/2}" loading="lazy"></span> is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients.

</p><p>In questions of prime number theory, such as <a href="Schinzel's_hypothesis_H" title="Schinzel's hypothesis H">Schinzel's hypothesis H</a> and the <a href="Bateman%E2%80%93Horn_conjecture" title="Bateman–Horn conjecture">Bateman–Horn conjecture</a>, it is a matter of basic importance to understand the case when <i>P</i> has no fixed prime divisor (this has been called <i>Bunyakovsky's property</i>, after <a href="Viktor_Bunyakovsky" title="Viktor Bunyakovsky">Viktor Bunyakovsky</a>). By writing <i>P</i> in terms of the binomial coefficients, we see the highest fixed prime divisor is also the highest prime <a href="Common_factor" class="mw-redirect" title="Common factor">common factor</a> of the coefficients in such a representation. So Bunyakovsky's property is equivalent to coprime coefficients.

</p><p>As an example, the pair of polynomials <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}+2}">

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<annotation encoding="application/x-tex">{\displaystyle n^{2}+2}</annotation>

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</math></span><img src="./b9c9a993589b2003937c9789a46c022f066ab024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.452ex; height:2.843ex;" alt="{\displaystyle n^{2}+2}" loading="lazy"></span> violates this condition at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=3}">

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<annotation encoding="application/x-tex">{\displaystyle p=3}</annotation>

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</math></span><img src="./b37be793795be56f61768c16dd72a1db0569fb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=3}" loading="lazy"></span>: for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> the product

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<annotation encoding="application/x-tex">{\displaystyle n(n^{2}+2)}</annotation>

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</math></span><img src="./94fe0d7e3d0629cc2554840cac92021515bc8f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.656ex; height:3.176ex;" alt="{\displaystyle n(n^{2}+2)}" loading="lazy"></span></dd></dl>

<p>is divisible by 3, which follows from the representation

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<annotation encoding="application/x-tex">{\displaystyle n(n^{2}+2)=6{\binom {n}{3}}+6{\binom {n}{2}}+3{\binom {n}{1}}}</annotation>

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</math></span><img src="./01500659ec82ef3272f267158a2c3d0cdde2dd55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.37ex; height:6.176ex;" alt="{\displaystyle n(n^{2}+2)=6{\binom {n}{3}}+6{\binom {n}{2}}+3{\binom {n}{1}}}" loading="lazy"></span></dd></dl>

<p>with respect to the binomial basis, where the highest common factor of the coefficients—hence the highest fixed divisor of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(n^{2}+2)}">

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<annotation encoding="application/x-tex">{\displaystyle n(n^{2}+2)}</annotation>

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</math></span><img src="./94fe0d7e3d0629cc2554840cac92021515bc8f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.656ex; height:3.176ex;" alt="{\displaystyle n(n^{2}+2)}" loading="lazy"></span>—is 3.

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<div class="mw-heading mw-heading2"><h2 id="Other_rings">Other rings</h2></div>

<p>Numerical polynomials can be defined over other rings and fields, in which case the integer-valued polynomials above are referred to as <b>classical numerical polynomials</b>.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>The <a href="Topological_K-theory" title="Topological K-theory">K-theory</a> of <a href="Classifying_space_for_U(n)" title="Classifying space for U(n)">BU(<i>n</i>)</a> is numerical (symmetric) polynomials.

</p><p>The <a href="Hilbert_polynomial" class="mw-redirect" title="Hilbert polynomial">Hilbert polynomial</a> of a polynomial ring in <i>k</i> + 1 variables is the numerical polynomial <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {t+k}{k}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\binom {t+k}{k}}}</annotation>

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</math></span><img src="./fb7836bd0793fd7eb61624277c6d97087914f82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.312ex; height:6.176ex;" alt="{\displaystyle {\binom {t+k}{k}}}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFJohnson2014" class="citation cs2">Johnson, Keith (2014), "Stable homotopy theory, formal group laws, and integer-valued polynomials", in Fontana, Marco; Frisch, Sophie; <a href="Sarah_Glaz" title="Sarah Glaz">Glaz, Sarah</a> (eds.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZZEpBAAAQBAJ&pg=PA213"><i>Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions</i></a>, Springer, pp. <span class="nowrap">213–</span>224, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781493909254</bdi></cite>. See in particular pp. 213–214.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Algebra">Algebra</h3></div>

<ul><li><cite id="CITEREFCahenChabert1997" class="citation cs2">Cahen, Paul-Jean; Chabert, Jean-Luc (1997), <i>Integer-valued polynomials</i>, Mathematical Surveys and Monographs, vol. 48, Providence, RI: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1421321">1421321</a></cite></li>

<li><cite id="CITEREFPólya1915" class="citation cs2 cs1-prop-foreign-lang-source"><a href="George_P%C3%B3lya" title="George Pólya">Pólya, George</a> (1915), "Über ganzwertige ganze Funktionen", <i>Palermo Rend.</i> (in German), <b>40</b>: <span class="nowrap">1–</span>16, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03014836">10.1007/BF03014836</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0009-725X">0009-725X</a>, <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:45.0655.02">45.0655.02</a></cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_topology">Algebraic topology</h3></div>

<ul><li><cite id="CITEREFBakerClarkeRaySchwartz1989" class="citation cs2">Baker, Andrew; Clarke, Francis; Ray, Nigel; Schwartz, Lionel (1989), "On the Kummer congruences and the stable homotopy of <i>BU</i>", <i><a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>, <b>316</b> (2): <span class="nowrap">385–</span>432, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2001355">10.2307/2001355</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2001355">2001355</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0942424">0942424</a></cite></li></ul>

<ul><li><cite id="CITEREFEkedahl2002" class="citation cs2">Ekedahl, Torsten (2002), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.hha/1139852462">"On minimal models in integral homotopy theory"</a>, <i><a href="Homology%2C_Homotopy_and_Applications" title="Homology, Homotopy and Applications">Homology, Homotopy and Applications</a></i>, <b>4</b> (2): <span class="nowrap">191–</span>218, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0107004">math/0107004</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fhha.2002.v4.n2.a9">10.4310/hha.2002.v4.n2.a9</a></span>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1918189">1918189</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1065.55003">1065.55003</a></cite></li></ul>

<ul><li><cite id="CITEREFElliott2006" class="citation journal cs1">Elliott, Jesse (2006). "Binomial rings, integer-valued polynomials, and λ-rings". <i><a href="Journal_of_Pure_and_Applied_Algebra" title="Journal of Pure and Applied Algebra">Journal of Pure and Applied Algebra</a></i>. <b>207</b> (1): <span class="nowrap">165–</span>185. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jpaa.2005.09.003">10.1016/j.jpaa.2005.09.003</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2244389">2244389</a>.</cite></li></ul>

<ul><li><cite id="CITEREFHubbuck1997" class="citation cs2">Hubbuck, John R. (1997), "Numerical forms", <i><a href="London_Mathematical_Society#Publications" title="London Mathematical Society">Journal of the London Mathematical Society</a></i>, Series 2, <b>55</b> (1): <span class="nowrap">65–</span>75, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2FS0024610796004395">10.1112/S0024610796004395</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1423286">1423286</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFNarkiewicz1995" class="citation book cs1">Narkiewicz, Władysław (1995). "Fully invariant sets for polynomial mappings". <i>Polynomial mappings</i>. Lecture Notes in Mathematics. Vol. 1600. Berlin: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. <span class="nowrap">67–</span>109. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0076896">10.1007/BFb0076896</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-59435-3</bdi>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-8434">0075-8434</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0829.11002">0829.11002</a>.</cite></li></ul></div><div><div class="zim-footer">

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