Additive Polynomial

Let k be a field of finite characteristic p. Then a polynomial P(x) in k[x] is said to be additive iff P(a)+P(b)=P(a+b) for {a,b,a+b} subset k. For example, P(x)=x^2+x+4 is additive for x in {1,2}, since


A more interesting class of additive polynomials known as absolutely additive polynomials are defined on an algebraic closure k^_ of k. For example, for any such k, tau_p(x)=x^p is an absolutely additive polynomial, since (p; j)=0 (mod p), for j=0, ..., p-1. The polynomial tau_p^i(x)=x^(p^i) is also absolutely additive.

Let the ring of polynomials spanned by linear combinations of tau_p^i be denoted k{tau_p}. If k!=F_p, then k{tau_p} is not commutative.

Not all additive polynomials are in k{tau_p}. In particular, if k is an infinite field, then a polynomial P(x) in k[x] is additive iff P(x) in k{tau_p}. For k be a finite field of characteristic p, the set of absolutely additive polynomials over k equals k{tau_p}, so the qualification "absolutely" can be dropped and the term "additive" alone can be used to refer to an element of k{tau_p}.

If p is a fixed power r=p^(k_0) and tau=tau_p^(k_0), then k{tau} is a ring of polynomials in tau. Moreover, if P(x) in k{tau}, then P(ax)=aP(x) for all a in F_r. In this case, P is said to be a F_r-linear polynomial.

The fundamental theorem of additive polynomials states that if P(x) in k[x] is a separable polynomial and {omega_1,...,omega_n} subset k is the set of its roots, then P(x) is additive iff if {omega_1,...,omega_n} is a subgroup.

It therefore follows as a corollary that such a polynomial P(x) is F_r-linear iff its roots form a F_r-vector subspace of k.

See also


This entry contributed by José Gallardo Alberni

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

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Additive Polynomial

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Alberni, José Gallardo. "Additive Polynomial." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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