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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 $\Bbb F_q[t],$ 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 $A,B,C \in \Bbb F_q[t]$ 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 $\Bbb F_2$ and $E + t^3$ over $F_4$ are decomposable. Thus, every polynomial in $\Bbb F_q[t]$ 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 $\Bbb F_q[t]$ 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 or rings 11T55 Arithmetic theory of polynomial rings over finite fields
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##### References:
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