Linearised polynomial

In mathematics, a linearised polynomial (or q- polynomial) is a polynomial for which the exponents of all the consituent monomials are powers of q and the coefficients come from some extension field of the finite field of order q.

We write a typical example as

$L(x)=\sum _{i=0}^{n}a_{i}x^{q^{i}},{\text{ where each }}a_{i}{\text{ is in }}GF(q^{m}){\text{ for some fixed positive integer }}m.$

Properties

• The map x → L(x) is a linear map on any field containing F_q
• The set of roots of L is an F_q-vector space and is closed under the q-Frobenius map
• Conversely, if U is any F_q-linear subspace of some finite field containing F_q, then the polynomial that vanishes exactly on U is a linearised polynomial.
• The set of linearised polynomials over a given field is closed under addition and composition of polynomials.

Associated polynomials

The polynomials L and

$l(x)=\sum _{i=0}^{n}a_{i}x^{i}\$

are associates.

References

• Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069.