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