DISCRIMINANT LOCI, GALOIS SYMMETRY, AND CM SPECIALIZATIONS IN A CUBIC FAMILY ARISING FROM AN ARC–DIFFERENCE ELIMINATION PROBLEM

A symbolic elimination problem motivated by an arc--difference system produces (i) a mixed-derivative identity in four variables and (ii) a degree-$12$ univariate elimination polynomial. We isolate the algebraic core of these outputs and interpret the resulting arithmetic geometry. First, we show that the mixed-derivative identity is equivalent, after a dimensionless normalization, to a one-parameter cubic equation \[ P_t(z)=t z^3-3t z^2+(3t+2)z+(1-t)=0 \] in the variable \(z=(q-s)^2/w^2\). The six radical branches observed in computer algebra output correspond to the three roots of \(P_t\) together with the independent sign choice \(q=s\pm w\sqrt{z}\). We compute \[ \Disc_z(P_t)=-t(243t+32) \] and parametrize the locus on which the discriminant is a square, yielding a rational conic. Second, we attach the genus-$1$ curve \(E_t:y^2=P_t(z)\) and obtain a short Weierstrass model \[ E_t:\ y^2=t x^3+2x+3, \qquad j(E_t)=\frac{55296}{32+243t}. \] In particular, under the natural positivity conditions inherited from the original system (forcing \(t>0\)), one has \(00}\). For completeness we also record the finite set of rational specializations \(t\in\QQ\) for which \(E_t\) has CM (equivalently, for which \(j(E_t)\) is a rational CM \(j\)-invariant). Third, we embed \(P_t\) into a two-parameter deformation \(P_{t,k}\) and identify the constant-CM subfamilies \(k=0\) (constant \(j=0\)) and \(k=-1\) (constant \(j=1728\)). Subsequent sections analyze the degree-$12$ elimination family, including a balanced specialization yielding a regular \(S_{11}\)-extension over \(\QQ(\tau)\), together with reproducible computations supplied as supplementary SageMath material.