Polynomial Pell equations with D(x)=x6+ax2+b

In this paper, we give a full characterization of solvability in [Formula: see text] of the polynomial Pell equations [Formula: see text] where a and b are rational, and we provide explicit formulae for all solutions. In particular, we prove that these equations have a nontrivial solution [Formula: see text] if and only if [Formula: see text] or [Formula: see text] or [Formula: see text] or [Formula: see text], where [Formula: see text]. We also show that if a and b are integers then the title equations have a nontrivial solution in [Formula: see text] if and only if [Formula: see text] or [Formula: see text]. It is worth noting that, to our knowledge, such full characterizations have so far been known only for polynomials [Formula: see text] of degree no greater than four. If [Formula: see text] has no repeated roots then these equations are connected with the arithmetic of the hyperelliptic curves.