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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 \(\mathbb F_q[t],\) 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 \(A,B,C \in \mathbb F_q[t]\) 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 \(\mathbb F_2\) and \(E + t^3\) over \(F_4\) are decomposable. Thus, every polynomial in \(\mathbb F_q[t]\) 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 \(\mathbb F_q[t]\) are \[ XY+Z,\quad X^2+X+YZ,\quad X^2+YZ. \] Observe that strict representations by the first and the last quadratic polynomials are trivial.

MSC:

11T06 Polynomials over finite fields
11T55 Arithmetic theory of polynomial rings over finite fields
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