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dditive Polynomial

المؤلف:  Goss, D

المصدر:  Basic Structures of Function Field Arithmetic. Berlin: Springer-Verlag

الجزء والصفحة:  pp. 1-33

17-1-2019

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dditive Polynomial

Let  be a field of finite characteristic . Then a polynomial  is said to be additive iff  for {a,b,a+b} subset k" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline5.gif" style="height:14px; width:90px" />. For example,  is additive for {1,2}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline7.gif" style="height:14px; width:54px" />, since

A more interesting class of additive polynomials known as absolutely additive polynomials are defined on an algebraic closure  of . For example, for any such ,  is an absolutely additive polynomial, since , for , ..., . The polynomial  is also absolutely additive.

Let the ring of polynomials spanned by linear combinations of  be denoted {tau_p}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline17.gif" style="height:20px; width:33px" />. If , then {tau_p}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline19.gif" style="height:20px; width:33px" /> is not commutative.

Not all additive polynomials are in {tau_p}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline20.gif" style="height:20px; width:33px" />. In particular, if  is an infinite field, then a polynomial  is additive iff {tau_p}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline23.gif" style="height:20px; width:75px" />. For  be a finite field of characteristic , the set of absolutely additive polynomials over  equals {tau_p}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline27.gif" style="height:20px; width:33px" />, so the qualification "absolutely" can be dropped and the term "additive" alone can be used to refer to an element of {tau_p}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline28.gif" style="height:20px; width:33px" />.

If  is a fixed power  and , then {tau}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline32.gif" style="height:14px; width:26px" /> is a ring of polynomials in . Moreover, if {tau}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline34.gif" style="height:14px; width:68px" />, then  for all . In this case,  is said to be a -linear polynomial.

The fundamental theorem of additive polynomials states that if  is a separable polynomial and {omega_1,...,omega_n} subset k" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline40.gif" style="height:14px; width:93px" /> is the set of its roots, then  is additive iff if {omega_1,...,omega_n}" src="http://mathworld.wolfram.com/images/equations/AdditivePolynomial/Inline42.gif" style="height:14px; width:72px" /> is a subgroup.

It therefore follows as a corollary that such a polynomial  is -linear iff its roots form a -vector subspace of .

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REFERENCES:

Goss, D. Basic Structures of Function Field Arithmetic. Berlin: Springer-Verlag, pp. 1-33, 1996.