Editing P-adic analysis

{{Short description|Branch of number theory}}
{{DISPLAYTITLE:''p''-adic analysis}}
[[Image:3-adic integers with dual colorings.svg|thumb|The 3-adic integers, with selected corresponding characters on their [[Pontryagin dual]] group]]
In [[mathematics]], '''''p''-adic analysis''' is a branch of [[number theory]] that studies functions of [[p-adic number|''p''-adic numbers]]. Along with the more classical fields of [[real analysis|real]] and [[complex analysis]], which deal, respectively, with functions on the [[real numbers|real]] and [[complex numbers|complex]] numbers, it belongs to the discipline of [[mathematical analysis]].

The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of [[locally compact group]]s ([[abstract harmonic analysis]]). The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest.

Applications of ''p''-adic analysis have mainly been in [[number theory]], where it has a significant role in [[diophantine geometry]] and [[diophantine approximation]]. Some applications have required the development of ''p''-adic [[functional analysis]] and [[spectral theory]]. In many ways ''p''-adic analysis is less subtle than [[classical analysis]], since the [[ultrametric inequality]] means, for example, that convergence of [[infinite series]] of ''p''-adic numbers is much simpler. [[Topological vector space]]s over ''p''-adic fields show distinctive features; for example aspects relating to [[convex set|convexity]] and the [[Hahn–Banach theorem]] are different.

==Important results==

===Ostrowski's theorem===
{{main article|Ostrowski's theorem}}
Ostrowski's theorem, due to [[Alexander Ostrowski]] (1916), states that every non-trivial [[absolute value (algebra)|absolute value]] on the [[rational number]]s '''Q''' is equivalent to either the usual real absolute value or a [[p-adic number|{{mvar|p}}-adic]] absolute value.<ref>{{cite book |last=Koblitz |first=Neal |authorlink=Neal Koblitz |title=P-adic numbers, p-adic analysis, and zeta-functions |series=Graduate Texts in Mathematics |year=1984 |volume=58 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-96017-3 |doi=10.1007/978-1-4612-1112-9 |edition=2nd  |page=3 |quote='''Theorem 1''' (Ostrowski). Every nontrivial norm ‖ ‖ on <math>\mathbb{Q}</math> is equivalent to {{math|{{abs|&nbsp;}}<sub>''p''</sub>}} for some prime {{mvar|p}} or for {{math|1=''p'' = ∞}}.}}</ref>

===Mahler's theorem===
{{main article|Mahler's theorem}}
'''Mahler's theorem''', introduced by [[Kurt Mahler]],<ref>{{Citation | last1=Mahler | first1=K. | title=An interpolation series for continuous functions of a p-adic variable | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846 | mr=0095821 | year=1958 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1958 | issue=199 | pages=23–34| doi=10.1515/crll.1958.199.23 | s2cid=199546556 }}
</ref> expresses continuous ''p''-adic functions in terms of polynomials.

In any [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] 0, one has the following result.  Let

:<math>(\Delta f)(x)=f(x+1)-f(x)</math>

be the forward [[difference operator]].  Then for [[polynomial function]]s ''f'' we have the [[Newton series]]:

:<math>f(x)=\sum_{k=0}^\infty (\Delta^k f)(0){x \choose k},</math>

where

:<math>{x \choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}</math>

is the ''k''th binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function ''f'' is a polynomial can be weakened, but it cannot be weakened all the way down to mere [[continuous function|continuity]].

Mahler proved the following result:

'''Mahler's theorem''': If ''f'' is a continuous [[p-adic number|''p''-adic]]-valued function on the ''p''-adic integers then the same identity holds.

===Hensel's lemma===
{{main article|Hensel's lemma}}
Hensel's lemma, also known as Hensel's lifting lemma, named after [[Kurt Hensel]], is a result in [[modular arithmetic]], stating that if a [[polynomial equation]] has a [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|simple root]] modulo a [[prime number]] {{math|''p''}}, then this root corresponds to a unique root of the same equation modulo any higher power of {{math|''p''}}, which can be found by iteratively "[[lift (mathematics)|lift]]ing" the solution modulo successive powers of {{math|''p''}}. More generally it is used as a generic name for analogues for [[completion (ring theory)|complete]] [[commutative ring]]s (including [[p-adic field|''p''-adic field]]s in particular) of the [[Newton method]] for solving equations. Since ''p''-adic analysis is in some ways simpler than [[real analysis]], there are relatively easy criteria guaranteeing a root of a polynomial.

To state the result, let <math>f(x)</math> be a [[polynomial]] with [[integer]] (or ''p''-adic integer) coefficients, and let ''m'',''k'' be positive integers such that ''m'' ≤ ''k''.  If ''r'' is an integer such that

:<math>f(r) \equiv 0 \pmod{p^k}</math> and <math>f'(r) \not\equiv 0 \pmod{p}</math>

then there exists an integer ''s'' such that

:<math>f(s) \equiv 0 \pmod{p^{k+m}}</math> and <math>r \equiv s \pmod{p^{k}}.</math>

Furthermore, this ''s'' is unique modulo ''p''<sup>''k''+m</sup>, and can be computed explicitly as

:<math>s = r + tp^k</math> where <math>t = - \frac{f(r)}{p^k} \cdot (f'(r)^{-1}).</math>

==Applications==

===Local–global principle===
{{main article|Local–global principle}}
[[Helmut Hasse]]'s local–global principle, also known as the Hasse principle, is the idea that one can find an [[diophantine equation|integer solution to an equation]] by using the [[Chinese remainder theorem]] to piece together solutions [[modular arithmetic|modulo]] powers of each different [[prime number]]. This is handled by examining the equation in the [[Completion (ring theory)|completions]] of the [[rational number]]s: the [[real number]]s and the [[p-adic number|''p''-adic numbers]]. A more formal version of the Hasse principle states that certain types of equations have a rational solution [[if and only if]] they have a solution in the [[real number]]s ''and'' in the ''p''-adic numbers for each prime ''p''.

==See also==
* [[P-adic exponential function|{{mvar|p}}-adic exponential function]]
* [[P-adic Teichmüller theory|{{mvar|p}}-adic Teichmüller theory]]
* [[Hypercomplex analysis]]
* [[P-adic quantum mechanics|{{mvar|p}}-adic quantum mechanics]]

==References==
{{reflist}}

==Further reading==
* {{cite book | last=Koblitz | first=Neal | authorlink=Neal Koblitz | title=p-adic analysis: a short course on recent work | series=London Mathematical Society Lecture Note Series | volume=46 | publisher=[[Cambridge University Press]] | year=1980 | isbn=0-521-28060-5 | zbl=0439.12011 }}  
* {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=Local Fields | series=London Mathematical Society Student Texts | volume=3 | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-31525-5 | zbl=0595.12006 }}
* {{cite journal |url=http://theory.cs.uni-bonn.de/Zope/English/csreports/report_1997/paper_85183/abstract.html |last1=Chistov |first1=Alexander |last2=Karpinski |first2=Marek |title=Complexity of Deciding Solvability of Polynomial Equations over p-adic Integers |journal=Univ. Of Bonn CS Reports 85183 |year=1997 |s2cid=120604553 }}
*{{cite journal|last1=Karpinski|first1=Marek|last2=van der Poorten|first2=Alf|last3=Shparlinski|first3=Igor|authorlink1=Marek Karpinski|year=2000|pages=309–317|journal=Theoretical Computer Science|volume=233|issue=1–2|title=Zero testing of p-adic and modular polynomials|doi=10.1016/S0304-3975(99)00133-4|citeseerx=10.1.1.131.6544}}
*A course in p-adic analysis, Alain Robert, Springer, 2000, {{isbn|978-0-387-98669-2}}
*Ultrametric Calculus: An Introduction to P-Adic Analysis, W. H. Schikhof, Cambridge University Press, 2007, {{isbn|978-0-521-03287-2}}
*P-adic Differential Equations, Kiran S. Kedlaya, Cambridge University Press, 2010, {{isbn|978-0-521-76879-5}}

{{Number systems}}

{{DEFAULTSORT:P-Adic Analysis}}
[[Category:Number theory]]
[[Category:Fields of mathematical analysis]]
[[Category:p-adic numbers]]