*(English)*Zbl 1127.11079

Serre proved that every polynomial of ${\mathbb{F}}_{q}\left[t\right],$ 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 Weilâ€™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 ${F}_{q}\left[t\right]$ are of the form (we say that they are decomposable):

where $A,B,C\in {\mathbb{F}}_{q}\left[t\right]$ satisfy the tight condition:

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 ${\mathbb{F}}_{q}\left[t\right]$ has a strict representation of the form:

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 ${\mathbb{F}}_{q}\left[t\right]$ are

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 |