P-derivation
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In mathematics, more specifically
differential algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A ...
, a ''p''-derivation (for ''p'' a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
) on a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
''R'', is a mapping from ''R'' to ''R'' that satisfies certain conditions outlined directly below. The notion of a ''p''-derivation is related to that of a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
in differential algebra.


Definition

Let ''p'' be a prime number. A ''p''-derivation or Buium derivative on a ring R is a map \delta:R\to R that satisfies the following "
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
": :\delta_p(ab) = \delta_p (a)b^p + a^p\delta_p (b) + p\delta_p (a)\delta_p (b) and "sum rule": :\delta_p(a+b) = \delta_p (a) + \delta_p(b) + \frac, as well as :\delta_p(1) = 0. Note that in the "sum rule" we are not really dividing by ''p'', since all the relevant
binomial coefficients In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
in the numerator are divisible by ''p'', so this definition applies in the case when R has ''p''- torsion.


Relation to Frobenius Endomorphisms

A map \sigma: R \to R is a lift of the
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...
provided \sigma(x) = x^p \pmod . An example of such a lift could come from the
Artin map The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...
. If (R, \delta) is a ring with a ''p''-derivation, then the map \sigma(x) := x^p + p\delta(x) defines a ring
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...
which is a lift of the Frobenius endomorphism. When the ring ''R'' is free the correspondence is a bijection.


Examples

* For R = \mathbb Z the unique ''p''-derivation is the map : \delta(x) = \frac. The quotient is well-defined because of Fermat's little theorem. * If ''R'' is any ''p''-torsion free ring and \sigma:R \to R is a lift of the Frobenius endomorphism then : \delta(x) = \frac defines a ''p''-derivation.


See also

*
Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of o ...
*
Arithmetic derivative In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. ...
*
Derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
*
Fermat quotient In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as= 3/ref> The smallest solutions of ''q'p''(''a'') ≡ 0 (mod ''p'') with ''a'' = ''n'' are: :2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, ...


References

* {{Citation, first=Alex, last=Buium, title=Arithmetic Differential Equations, year=1989, publisher=Springer-Verlag, isbn=0-8218-3862-8, series=Mathematical Surveys and Monographs.


External links


Project Euclid
Differential algebra Generalizations of the derivative