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Additive Polynomial

An additive polynomial over a field k of positive field characteristic p is a polynomial P in k[x] satisfying P(a+b)=P(a)+P(b) for all a,b in k. The stronger condition that this holds over an algebraic closure of k is called absolute additivity (Goss 1996). Equivalently, it is the polynomial identity

P(X+Y)=P(X)+P(Y)
(1)

in k[X,Y]. Some authors use "additive polynomial" for this stronger condition (Kuhlmann 2011, pp. 301-302).

Absolute additivity is equivalent to having the form

P(x)=sum_(i=0)^mc_ix^(p^i),
(2)

where c_i in k and m is a nonnegative integer. In particular, the Frobenius map tau_p(x)=x^p and its iterates tau_p^i(x)=x^(p^i) are absolutely additive. The binomial theorem gives this property since (p; j) is divisible by p for 1<=j<=p-1 (Kuhlmann 2011, p. 302).

Over an infinite field, additivity on k already implies the polynomial identity. Over a finite field, distinct polynomials can induce the same polynomial function, so the two conditions differ. For example, over F_p, P(x)=x^(p+1)-x^2 induces the zero function, but it is not absolutely additive (Kuhlmann 2011, p. 301).

The absolutely additive polynomials form a ring k{tau_p} under addition and composition, rather than ordinary polynomial multiplication. A scalar c in k acts as the polynomial x|->cx, and

tau_pc=c^ptau_p.
(3)

Thus this ring is commutative iff k=F_p (Kuhlmann 2011, p. 303).

More generally, suppose k contains the finite field F_r, where r=p^s and s is a positive integer. The polynomials in k{tau_p^s} are precisely those of the form sum_(i)c_ix^(r^i). They satisfy P(ax)=aP(x) for every a in F_r over an algebraic closure, in addition to absolute additivity, and are called F_r-linear polynomials (Gow and McGuire 2025, p. 5159).

For a nonzero polynomial P in k[x] with no multiple roots, absolute additivity is equivalent to its full set of roots in an algebraic closure being a subgroup of the additive group of that field (Kuhlmann 2011, p. 302). In the absolutely additive form above, P^'(x)=c_0, so it has no multiple root iff c_0!=0. Under the same assumption F_r subset= k, a nonzero polynomial with no multiple roots is F_r-linear iff its full set of roots in an algebraic closure is an F_r-vector subspace (Gow and McGuire 2025, pp. 5160-5161). Here the no-multiple-root condition is essential and uses the rootwise convention for a separable polynomial.

See also

Frobenius Map, Polynomial, Polynomial Function, Separable Polynomial

Portions of this entry contributed by José Gallardo Alberni

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References

Goss, D. Basic Structures of Function Field Arithmetic. Berlin, Germany: Springer-Verlag, pp. 1-33, 1996.Gow, R. and McGuire, G. "Linearization of Polynomials in Prime Characteristic, with Applications to the Golay Code and Steiner System." Des. Codes Cryptogr. 93, 5159-5177, 2025. https://doi.org/10.1007/s10623-025-01724-7.Kuhlmann, F.-V. "Additive Polynomials." Ch. 12 of a book on valuation theory in preparation. Version of Jul. 14, 2011, pp. 301-303. https://www.fvkuhlmann.de/bookch12.pdf.

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Additive Polynomial

Cite this as:

Weisstein, Eric W., with contributions by José Gallardo Alberni. "Additive Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AdditivePolynomial.html

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