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

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

This special class of polynomials is important from both a theoretical and an applications viewpoint.^[1] The highly structured nature of their roots makes these roots easy to determine.

Properties

• The map x → L(x) is a linear map over 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.

Symbolic multiplication and division

In general, the product of two linearised polynomials will not be a linearized polynomial, but since the composition of two linearised polynomials results in a linearised polynomial, composition may be used as a replacement for multiplication and, for this reason, composition is often called symbolic multiplication in this setting. Notationally, if L₁(x) and L₂(x) are linearised polynomials we define

${\displaystyle L_{1}(x)\otimes L_{2}(x)=L_{1}(L_{2}(x))}$ 

when this point of view is being taken.

In the special case that the linearised polynomials are defined over F_q, it can be shown that, as an operation, symbolic multiplication is commutative, associative and distributes over ordinary addition.^[2] Also, in this special case, we can define the operation of symbolic division. If L(x) and L₁(x) are linearised polynomials over F_q, we say that L₁(x) symbolically divides L(x) if there exists a linearised polynomial L₂(x) over F_q for which:

${\displaystyle L(x)=L_{1}(x)\otimes L_{2}(x).}$ 

Associated polynomials

The polynomials L(x) and

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

are q - associates (note: the exponents "qⁱ " of L(x) have been replaced by "i" in l(x)). More specifically, l(x} is called the conventional q-associate of L(x), and L(x) is the linearised q-associate of l(x).

If L₁(x) and L₂(x) are linearised polynomials over F_q with conventional q-associates l₁(x) and l₂(x) respectively, then L₁(x) symbolically divides L₂(x) if and only if l₁(x) divides l₂(x).^[3] Furthermore, L₁(x) divides L₂(x) in the ordinary sense in this case.^[4]

Notes

1. Lidl & Niederreiter 1983, pg.107 (first edition) harvnb error: no target: CITEREFLidlNiederreiter1983 (help)
2. Lidl & Niederreiter 1983, pg. 115 (first edition) harvnb error: no target: CITEREFLidlNiederreiter1983 (help)
3. Lidl & Niederreiter 1983, pg. 115 (first edition) Corollary 3.60 harvnb error: no target: CITEREFLidlNiederreiter1983 (help)
4. Lidl & Neiderreiter 1983, pg. 116 (first edition) Theorem 3.62 harvnb error: no target: CITEREFLidlNeiderreiter1983 (help)

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.