In mathematics, a linearised polynomial is a polynomial for which the exponents of all the constituent monomials are powers of q and the coefficientscome from some extension field of the finite field of order q. We write a typical example as This special class of polynomials is important from both a theoretical and an applications viewpoint. The highly structured nature of their roots makes these roots easy to determine.
Properties
The map x → L is a linear map over any field containing Fq
The set of roots of L is an Fq-vector space and is closed under the q-Frobenius map
Conversely, if U is any Fq-linear subspace of some finite field containing Fq, 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.
If L is a nonzero linearised polynomial over with all of its roots lying inthe field an extension field of, then each root of L has the same multiplicity, which is either 1, or a positive power of q.
Symbolic multiplication
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 and L2 are linearised polynomials we define when this point of view is being taken.
Associated polynomials
The polynomials L and are q - associates have been replaced by "i" in l). More specifically, ''l(x
q-polynomials over '''F'''''q''
Linearised polynomials with coefficients in Fq have additional properties which make it possible to define symbolic division, symbolic reducibility and symbolic factorization. Two important examples of this type of linearised polynomial are the Frobenius automorphism and the trace function. In this special case it can be shown that, as an operation, symbolic multiplication is commutative, associative and distributes over ordinary addition. Also, in this special case, we can define the operation of symbolic division. If L and L1 are linearised polynomials over Fq, we say that L1symbolically dividesL if there exists a linearised polynomial L2 over Fq for which: If L1 and L2 are linearised polynomials over Fq with conventional q-associates l1 and l2 respectively, then L1 symbolically divides L2if and only ifl1 divides l2. Furthermore, L1 divides L2 in the ordinary sense in this case. A linearised polynomial L over Fq of degree > 1 is symbolically irreducible over Fq if the only symbolic decompositions with Li over Fq are those for which one of the factors has degree 1. Note that a symbolically irreducible polynomial is always reducible in the ordinary sense since any linearised polynomial of degree > 1 has the nontrivial factor x. A linearised polynomial L over Fq is symbolically irreducible if and only if its conventional q-associate l is irreducible over Fq. Every q-polynomial L over Fq of degree > 1 has a symbolic factorization into symbolically irreducible polynomials over Fq and this factorization is essentially unique For example, consider the 2-polynomial L = x16 + x8 + x2 + x over F2 and its conventional 2-associate l = x4 + x3 + x + 1. The factorization into irreducibles of l = 2 in F2, gives the symbolic factorization
Affine polynomials
Let L be a linearised polynomial over. A polynomial of the form is an affine polynomial over. Theorem: If A is a nonzero affine polynomial over with all of its roots lying in the field an extension field of, then each root of A has the same multiplicity, which is either 1, or a positive power of q.
