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Representations of polynomials over finite fields of characteristic two as A 2 +A+BC+D 3 . (English) Zbl 1127.11079

Serre proved that every polynomial of 𝔽 q [t], with q odd (with a small number of exceptions when q=3,) is a strict sum of three squares. The authors prove by using the same method, (apply Weil’s theorem for an appropriate curve) that for even q all (but a finite number of polynomials when q<8,, all explicitly stated in the paper) polynomials P of F q [t] are of the form (we say that they are decomposable):

P=A 2 +A+BC

where A,B,C𝔽 q [t] satisfy the tight condition:

max(deg(A 2 ),deg(B 2 ),deg(C 2 ))<deg(P)+2·

The exceptions E are well behaved in the sense that it is easy to prove that for all of them E+1 3 over 𝔽 2 and E+t 3 over F 4 are decomposable. Thus, every polynomial in 𝔽 q [t] has a strict representation of the form:

P=A 2 +A+BC+D 3 ·

It is also proved that for every even q the only quadratic polynomials in three variables X,Y,Z that represent strictly all (but a finite number) of polynomials of 𝔽 q [t] are

XY+Z,X 2 +X+YZ,X 2 +YZ·

Observe that strict representations by the first and the last quadratic polynomials are trivial.

11T06Polynomials over finite fields or rings
11T55Arithmetic theory of polynomial rings over finite fields