The valuative capacity of the set of sums of d-th powers

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• 作者：Marie-André ; e B. Langlois ^{mblanglois@dal.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Integer valued polynomials ; Valuative capacity ; p-Orderings ; Sums of powers
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：155-168
• 全文大小：346 K

文摘

If m>Em> is a subset of the integers then the m>nm>-th characteristic ideal of m>E  m> is the fractional ideal of d="mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301895&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301895&_rdoc=1&_issn=0022314X&md5=2ecfafd670f16dfd4a8c79847d161ec1" title="Click to view the MathML source">ZmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi mathvariant="double-struck">Zmi>math> consisting of 0 and the leading coefficients of the polynomials in d="mmlsi155" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301895&_mathId=si155.gif&_user=111111111&_pii=S0022314X16301895&_rdoc=1&_issn=0022314X&md5=322d6da420d58e5723b4f41a1bdd3a60" title="Click to view the MathML source">Q[x]mathContainer hidden">mathCode"><math altimg="si155.gif" overflow="scroll"><mi mathvariant="double-struck">Qmi><mo stretchy="false">[mo><mi>xmi><mo stretchy="false">]mo>math> of degree no more than m>nm> which are integer valued on m>Em>. For m>p  m> a prime the characteristic sequence of d="mmlsi3" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301895&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301895&_rdoc=1&_issn=0022314X&md5=f092be2be8c66ec117d0a6dccdeb6716" title="Click to view the MathML source">m>Intm>(E,Z)mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><mrow><mi mathvariant="italic">Intmi>mrow><mo stretchy="false">(mo><mi>Emi><mo>,mo><mi mathvariant="double-struck">Zmi><mo stretchy="false">)mo>math> is the sequence d="mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301895&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301895&_rdoc=1&_issn=0022314X&md5=a90d713981520eccce2184cca9a2b4d8" title="Click to view the MathML source">α_E(n)mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msub><mrow><mi>αmi>mrow><mrow><mi>Emi>mrow>msub><mo stretchy="false">(mo><mi>nmi><mo stretchy="false">)mo>math> of negatives of the m>p  m>-adic valuations of these ideals. The asymptotic limit d="mmlsi5" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301895&_mathId=si5.gif&_user=111111111&_pii=S0022314X16301895&_rdoc=1&_issn=0022314X&md5=58a4f46b1352ecc3cbfdcf4532f1b7cb">mg class="imgLazyJSB inlineImage" height="22" width="110" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301895-si5.gif">mathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><msub><mrow><mi mathvariant="normal">limmi>mrow><mrow><mi>nmi><mo stretchy="false">→mo><mo>∞mo>mrow>msub><mo>⁡mo><mfrac><mrow><msub><mrow><mi>αmi>mrow><mrow><mi>Emi><mo>,mo><mi>pmi>mrow>msub><mo stretchy="false">(mo><mi>nmi><mo stretchy="false">)mo>mrow><mi>nmi>mfrac>math> of this sequence, called the valuative capacity of m>Em>, gives information about the geometry of m>Em>. We compute these valuative capacities for the sets m>E  m> of sums of d="mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301895&_mathId=si6.gif&_user=111111111&_pii=S0022314X16301895&_rdoc=1&_issn=0022314X&md5=5346672e5f9ea81490b8a1685b66017e" title="Click to view the MathML source">ℓ≥2mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>ℓmi><mo>≥mo><mn>2mn>math> integers to the power of m>dm>, by observing the m>pm>-adic closure of these sets.