Free Rational Parametrizations of Châtelet Surfaces, with Applications to Three Equations of Grechuk

Let k be a number field, a ∈ k× a non-square, and f ∈ k[x] separable of degree 3 or 4. We give explicit rational maps from open subsets of affine 5-space over k covering all non-degenerate rational points of y² − az² = f(x). There is one map per realized type in an explicit quotient of an étale norm group. The parameters satisfy only denominator non-vanishing conditions. A universal witness lemma, proved using complementary Vandermonde minors, makes the formulas exhaustive on every relevant torsor fibre. Normalization of a coefficient and a Laurent-monomial argument reduce each chart to five free parameters. We prove directly that type is invariant along rational curves, including affine chains through degenerate points; consequently the number of charts is minimal, for both cubic and quartic f. The list of realized types is computable, using local norm conditions and Molyakov’s published Hasse principle for intersections of two quadrics in projective 7-space. Given a point of a torsor fibre, at most 4845 exact trials suffice to find a nonzero chart determinant; for a canonical chart and a prescribed target, at most 135751 trials suffice to recover all five parameters. We apply the construction to three equations of Grechuk: two five-parameter charts suffice for each cubic equation, and one suffices for y² − x²y + z² + 1 = 0. For the latter we also give a complete fibre criterion, explicit inverses on fibres, and a denominator obstruction. No optimality is claimed for the number five of parameters.