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词条 Davenport constant
释义

  1. Example

  2. Properties

  3. Applications

  4. Variants

  5. References

  6. External links

{{more citations needed|date=May 2013}}

In mathematics, the Davenport constant {{Math|D(G)}} is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite abelian group {{Mvar|G}} is defined as the smallest number, such that every sequence of elements of that length contains a non-empty sub-sequence adding up to 0. In symbols, this is[1]

.

Example

  • The Davenport constant for the cyclic group {{Math|1=G = 𝕫/(n)}} is {{Mvar|n}}. To see this note that the sequence of a fixed generator, repeated {{Math|n-1}} times, contains no sub-sequence with sum {{Math|0}}. Thus {{Math|D(G) ≥ n}}. On the other hand, if is an arbitrary sequence, then two of the sums in the sequence are equal. The difference of these two sums also gives a sub-sequence with sum {{Math|0}}.{{Sfn|Geroldinger|2009|p=24}}

Properties

  • Consider a finite abelian group {{Math|1=G=⊕{{sub|i}}C{{sub|d{{sub|i}}}}}}, where the {{Math|d{{sub|1}}{{!}}d{{sub|2}}{{!}}…{{!}}d{{sub|r}}}} are invariant factors. Then

.

The lower bound is proved by noting that the sequence "{{Math|d{{sub|1}}-1}} copies of {{Math|(1, 0, …, 0)}}, {{Math|d{{sub|2}}-1}} copies of {{Math|(0, 1, …, 0)}}, etc." contains no subsequence with sum {{Math|0}}.[2]

  • {{Math|1=D=M}} for p-groups or for {{Math|r∈{1,2}}}.{{Clarification needed|reason=What is r?|date=August 2018}}
  • {{Math|1=D=M}} for certain groups including all groups of the form {{Math|C{{sub|2}}⊕C{{sub|2n}}⊕C{{sub|2nm}}}} and {{Math|C{{sub|3}}⊕C{{sub|3n}}⊕C{{sub|3nm}}}}.
  • There are infinitely many examples with {{Mvar|r}} at least {{Math|4}} where {{Mvar|D}} does not equal {{Mvar|M}}; it is not known whether there are any with {{Math|1=r = 3}}.[2]
  • Let be the exponent of {{Mvar|G}}. Then[4]

.

Applications

The original motivation for studying Davenport's constant was the problem of non-unique factorization in number fields. Let be the ring of integers in a number field, {{Mvar|G}} its class group. Then every element , which factors into at least {{Math|D(G)}} non-trivial ideals, is properly divisible by an element of . This observation implies that Davenport's constant determines by how much the lengths of different factorization of some element in can differ.[3]{{Citation needed|date=August 2018}}

The upper bound mentioned above plays an important role in Ahlford, Granville and Pomerance's proof of the existence of infinitely many Carmichael numbers.[4]

Variants

Olson's constant {{Math|O(G)}} uses the same definition, but requires the elements of to be pairwise different.[5]

  • Balandraud proved that {{Math|O(C{{sub|p}})}} equals the smallest {{Mvar|k}}, such that .
  • For {{Math|p>6000}}, we have

.

On the other hand, if {{Math|1=G=C{{su|b=p|p=r}}}} with {{Math|r ≥ p}}, then Olson's constant equals the Davenport constant.[6]

References

1. ^{{cite book|title=Combinatorial number theory and additive group theory|last=Geroldinger|first=Alfred|publisher=Birkhäuser|others=Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Sólymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse)|year=2009|isbn=978-3-7643-8961-1|editor1-last=Geroldinger|editor1-first=Alfred|series=Advanced Courses in Mathematics CRM Barcelona|location=Basel|pages=1–86|chapter=Additive group theory and non-unique factorizations|zbl=1221.20045|ref=harv|editor2-last=Ruzsa|editor2-first=Imre Z.|editor2-link=Imre Z. Ruzsa|doi=10.1007/978-3-7643-8962-8}}
2. ^1 {{cite book|title=Additive combinatorics|last1=Bhowmik|first1=Gautami|last2=Schlage-Puchta|first2=Jan-Christoph|publisher=American Mathematical Society|year=2007|isbn=978-0-8218-4351-2|editor1-last=Granville|editor1-first=Andrew|editor1-link=Andrew Granville|series=CRM Proceedings and Lecture Notes|volume=43|location=Providence, RI|pages=307–326|chapter=Davenport's constant for groups of the form {{math|𝕫3⊕𝕫3⊕𝕫3d}}|zbl=1173.11012|editor2-last=Nathanson|editor2-first=Melvyn B.|editor3-last=Solymosi|editor3-first=József|editor3-link= József Solymosi |chapter-url=http://math.univ-lille1.fr/~bhowmik/files/333d.pdf}}
3. ^{{Cite journal|date=1969-01-01|title=A combinatorial problem on finite Abelian groups, I|url=https://www.sciencedirect.com/science/article/pii/0022314X69900213|journal=Journal of Number Theory|language=en|volume=1|issue=1|pages=8–10|doi=10.1016/0022-314X(69)90021-3|issn=0022-314X|last1=Olson|first1=John E.}}
4. ^1 {{cite journal|author=W. R. Alford|author2=Andrew Granville|author3=Carl Pomerance|year=1994|title=There are Infinitely Many Carmichael Numbers|url=http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf|journal=Annals of Mathematics|volume=139|issue=3|pages=703–722|doi=10.2307/2118576|jstor=2118576}}
5. ^{{Cite journal|date=2012-01-01|title=A characterization of incomplete sequences in vector spaces|journal=Journal of Combinatorial Theory, Series A|language=en|volume=119|issue=1|pages=33–41|doi=10.1016/j.jcta.2011.06.012|issn=0097-3165|last1=Nguyen|first1=Hoi H.|last2=Vu|first2=Van H.}}
6. ^{{Cite journal|last=Ordaz|first=Oscar|last2=Philipp|first2=Andreas|last3=Santos|first3=Irene|last4=Schmidt|first4=Wolfgang A.|date=2011|title=On the Olson and the Strong Davenport constants|url=http://www.numdam.org/article/JTNB_2011__23_3_715_0.pdf|journal=Journal de Théorie de Nombres de Bordeaux|volume=23|issue=3|pages=715–750|via=NUMDAM|doi=10.5802/jtnb.784}}
  • {{cite book|first=Melvyn B.|last=Nathanson|title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets|volume=165|series=Graduate Texts in Mathematics|publisher=Springer-Verlag|year=1996|isbn=978-0-387-94655-9|zbl=0859.11003}}

External links

  • {{springer|title=Davenport constant|id=p/d110010}}
  • {{MathWorld | title=Davenport Constant | id=DavenportConstant | author=Hutzler, Nick }}

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