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 Formula: see text and Formula: see text 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 Formula: see text and Formula: see text 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.