p-adic analysis

<h2 id="important-results">Important results</h2>
<h3>Ostrowski's theorem</h3>
Main article: <a href="/facts/Ostrowski%27s_theorem/VWs7mDQt">Ostrowski's theorem</a>
Ostrowski's theorem, due to <a href="/facts/Alexander_Ostrowski/r45sj25p">Alexander Ostrowski</a> (1916), states that every non-trivial <a href="/facts/Absolute_value_(algebra)/UUncm7y0">absolute value</a> on the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> Q is equivalent to either the usual real absolute value or a <a href="/facts/P-adic_number/SDgDbkLF">p-adic</a> absolute value.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h3>Mahler's theorem</h3>
Main article: <a href="/facts/Mahler%27s_theorem/cldgGtbS">Mahler's theorem</a>
Mahler's theorem, introduced by <a href="/facts/Kurt_Mahler/8VfW4MWQ">Kurt Mahler</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> expresses continuous p-adic functions in terms of polynomials.
In any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> 0, one has the following result. Let

(
        Δ
        f
        )
        (
        x
        )
        =
        f
        (
        x
        +
        1
        )
        −
        f
        (
        x
        )
      
    
    {\displaystyle (\Delta f)(x)=f(x+1)-f(x)}

be the forward <a href="/facts/Difference_operator/aY1txFhj">difference operator</a>. Then for <a href="/facts/Polynomial_function/Lzak8VVx">polynomial functions</a> f we have the <a href="/facts/Newton_series/aY1txFhj">Newton series</a>:

f
        (
        x
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        (
        
          Δ
          
            k
          
        
        f
        )
        (
        0
        )
        
          
            
              (
            
            
              x
              k
            
            
              )
            
          
        
        ,
      
    
    {\displaystyle f(x)=\sum _{k=0}^{\infty }(\Delta ^{k}f)(0){x \choose k},}

where

(
            
            
              x
              k
            
            
              )
            
          
        
        =
        
          
            
              x
              (
              x
              −
              1
              )
              (
              x
              −
              2
              )
              ⋯
              (
              x
              −
              k
              +
              1
              )
            
            
              k
              !
            
          
        
      
    
    {\displaystyle {x \choose k}={\frac {x(x-1)(x-2)\cdots (x-k+1)}{k!}}}

is the kth 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 <a href="/facts/Continuous_function/sKbl02pB">continuity</a>.
Mahler proved the following result:
Mahler's theorem: If f is a continuous <a href="/facts/P-adic_number/SDgDbkLF">p-adic</a>-valued function on the p-adic integers then the same identity holds.

<h3>Hensel's lemma</h3>
Main article: <a href="/facts/Hensel%27s_lemma/CPCADITS">Hensel's lemma</a>
Hensel's lemma, also known as Hensel's lifting lemma, named after <a href="/facts/Kurt_Hensel/xoKnoBfv">Kurt Hensel</a>, is a result in <a href="/facts/Modular_arithmetic/0wtRa5dU">modular arithmetic</a>, stating that if a <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equation</a> has a <a href="/facts/Multiplicity_(mathematics)/dndUyetA">simple root</a> modulo a <a href="/facts/Prime_number/L20FLkgn">prime number</a> p, then this root corresponds to a unique root of the same equation modulo any higher power of p, which can be found by iteratively "<a href="/facts/Lift_(mathematics)/WCqXTXVI">lifting</a>" the solution modulo successive powers of p. More generally it is used as a generic name for analogues for <a href="/facts/Completion_(ring_theory)/Q0vjA9JI">complete</a> <a href="/facts/Commutative_ring/QS1r2ovR">commutative rings</a> (including <a href="/facts/P-adic_field/SDgDbkLF">p-adic fields</a> in particular) of the <a href="/facts/Newton_method/VlRhI2FL">Newton method</a> for solving equations. Since p-adic analysis is in some ways simpler than <a href="/facts/Real_analysis/12WjlXCW">real analysis</a>, there are relatively easy criteria guaranteeing a root of a polynomial.
To state the result, let 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 be a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> with <a href="/facts/Integer/PUwwrawx">integer</a> (or p-adic integer) coefficients, and let m,k be positive integers such that m ≤ k. If r is an integer such that

f
        (
        r
        )
        ≡
        0
        
          
          (
          mod
          
          
            p
            
              k
            
          
          )
        
      
    
    {\displaystyle f(r)\equiv 0{\pmod {p^{k}}}}
  
 and 
  
    
      
        
          f
          ′
        
        (
        r
        )
        ≢
        0
        
          
          (
          mod
          
          p
          )
        
      
    
    {\displaystyle f'(r)\not \equiv 0{\pmod {p}}}

then there exists an integer s such that

f
        (
        s
        )
        ≡
        0
        
          
          (
          mod
          
          
            p
            
              k
              +
              m
            
          
          )
        
      
    
    {\displaystyle f(s)\equiv 0{\pmod {p^{k+m}}}}
  
 and 
  
    
      
        r
        ≡
        s
        
          
          (
          mod
          
          
            p
            
              k
            
          
          )
        
        .
      
    
    {\displaystyle r\equiv s{\pmod {p^{k}}}.}

Furthermore, this s is unique modulo pk+m, and can be computed explicitly as

s
        =
        r
        +
        t
        
          p
          
            k
          
        
      
    
    {\displaystyle s=r+tp^{k}}
  
 where 
  
    
      
        t
        =
        −
        
          
            
              f
              (
              r
              )
            
            
              p
              
                k
              
            
          
        
        ⋅
        (
        
          f
          ′
        
        (
        r
        
          )
          
            −
            1
          
        
        )
        .
      
    
    {\displaystyle t=-{\frac {f(r)}{p^{k}}}\cdot (f'(r)^{-1}).}

<h2 id="applications">Applications</h2>
<h3>Local–global principle</h3>
Main article: <a href="/facts/Local%E2%80%93global_principle/B6el223t">Local–global principle</a>
<a href="/facts/Helmut_Hasse/WFtCX2jI">Helmut Hasse</a>'s local–global principle, also known as the Hasse principle, is the idea that one can find an <a href="/facts/Diophantine_equation/mYaHp7oX">integer solution to an equation</a> by using the <a href="/facts/Chinese_remainder_theorem/7wwFIkuV">Chinese remainder theorem</a> to piece together solutions <a href="/facts/Modular_arithmetic/0wtRa5dU">modulo</a> powers of each different <a href="/facts/Prime_number/L20FLkgn">prime number</a>. This is handled by examining the equation in the <a href="/facts/Completion_(ring_theory)/Q0vjA9JI">completions</a> of the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>: the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and the <a href="/facts/P-adic_number/SDgDbkLF">p-adic numbers</a>. A more formal version of the Hasse principle states that certain types of equations have a rational solution <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> they have a solution in the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and in the p-adic numbers for each prime p.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/P-adic_exponential_function/T5wK0u3g">p-adic exponential function</a></li>
<li><a href="/facts/P-adic_Teichm%C3%BCller_theory/4LNya5gN">p-adic Teichmüller theory</a></li>
<li><a href="/facts/Hypercomplex_analysis/V996HnBp">Hypercomplex analysis</a></li>
<li><a href="/facts/P-adic_quantum_mechanics/WKzHv4VA">p-adic quantum mechanics</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Neal_Koblitz/TebdlGqw">Koblitz, Neal</a> (1980). p-adic analysis: a short course on recent work. London Mathematical Society Lecture Note Series. Vol. 46. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-28060-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0439.12011">0439.12011</a>.</li>
<li><a href="/facts/J._W._S._Cassels/AC33eVoA">Cassels, J.W.S.</a> (1986). Local Fields. London Mathematical Society Student Texts. Vol. 3. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-31525-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0595.12006">0595.12006</a>.</li>
<li>Chistov, Alexander; Karpinski, Marek (1997). <a href="http://theory.cs.uni-bonn.de/Zope/English/csreports/report_1997/paper_85183/abstract.html">"Complexity of Deciding Solvability of Polynomial Equations over p-adic Integers"</a>. Univ. Of Bonn CS Reports 85183. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120604553">120604553</a>.</li>
<li><a href="/facts/Marek_Karpinski/XPxNRbPA">Karpinski, Marek</a>; van der Poorten, Alf; Shparlinski, Igor (2000). "Zero testing of p-adic and modular polynomials". Theoretical Computer Science. 233 (1–2): 309–317. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.6544">10.1.1.131.6544</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0304-3975%2899%2900133-4">10.1016/S0304-3975(99)00133-4</a>.</li>
<li>A course in p-adic analysis, Alain Robert, Springer, 2000, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98669-2</li>
<li>Ultrametric Calculus: An Introduction to P-Adic Analysis, W. H. Schikhof, Cambridge University Press, 2007, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-03287-2</li>
<li>P-adic Differential Equations, Kiran S. Kedlaya, Cambridge University Press, 2010, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-76879-5</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions. Graduate Texts in Mathematics. Vol. 58 (2nd ed.). New York: Springer-Verlag. p. 3. doi:10.1007/978-1-4612-1112-9. ISBN 978-0-387-96017-3. Theorem 1 (Ostrowski). Every nontrivial norm ‖ ‖ on 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 is equivalent to | |p for some prime p or for p = ∞. <a href="978-0-387-96017-3" target="_blank">978-0-387-96017-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Mahler, K. (1958), "An interpolation series for continuous functions of a p-adic variable", Journal für die reine und angewandte Mathematik, 1958 (199): 23–34, doi:10.1515/crll.1958.199.23, ISSN 0075-4102, MR 0095821, S2CID 199546556 <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846" target="_blank">http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

p-adic analysis open-in-new

p-adic analysis