Decidability of rational solvability for multivariate polynomial equations

Determine the decidability of the problem that, given a multivariate polynomial equation with coefficients in the rational numbers, decides whether the equation has a solution in the rational numbers. Establish whether a general algorithm exists to decide rational solvability for arbitrary multivariate polynomial equations over Q.

Background

The paper reduces loop synthesis with polynomial invariants to solving systems of polynomial equations whose solutions encode all admissible loop update maps. Since the implementation targets exact, finitely representable programs, the focus is on solutions with rational coefficients.

The authors note that, unlike the algebraic and real cases where general decision procedures exist (via Hilbert's Nullstellensatz and Tarski–Seidenberg, respectively), the existence of a general decision procedure over the rationals is a longstanding open problem in number theory. This limitation motivates their use of structural decomposition, numerical homotopy continuation, and SMT solvers as practical but incomplete strategies.

References

Unlike solutions in C (by Hilbert's Nullstellensatz , see e.g., Chap. 4, \S 1) or R (via Tarski-Seidenberg’s theorem , see e.g., ), determining whether a multivariate polynomial equation has a rational solution is a major open problem in number theory .

— Beyond Affine Loops: A Geometric Approach to Program Synthesis  (2505.00620 - Bayarmagnai et al., 1 May 2025) in Section 4, Polynomial system solving

In the case of a smooth plane cubic curve $\mathfrak{C}$, problem (P1) is still open, that is, we do not have a general technique to establish whether DiophCubic admits a rational point or not.

DiophCubic:

ax3+bx2y+cxy2+dy3+ex2+fxy+gy2 +hx+iy+j=0,a,…,j∈Z,  (a,b,c,d)≠(0,0,0,0)\begin{split} ax^{3} + bx^{2}y + cxy^{2} + dy^{3} + ex^{2} + fxy + gy^{2} \, & + hx + iy + j = 0, \\ & a, \ldots, j \in \Z, \; (a, b, c, d) \neq (0, 0, 0, 0) \end{split}

— Introduction to the Birch and Swinnerton-Dyer Conjecture  (2608.25975 - Ruffino, 26 Aug 2026) in Section 2, Section 2.3, subsection “Smooth cubics with a rational point”