Hilbert's Nullstellensatz is in the Counting Hierarchy

Abstract: We show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that the number of solutions to a system of equations can be computed in polynomial time with oracle access to the counting hierarchy. Our results hold in particular for polynomials with coefficients in either the rational numbers or a finite field. Previously, the best-known bounds on the complexities of these problems were PSPACE and FPSPACE, respectively. Our main technical contribution is the construction of a uniform family of constant-depth arithmetic circuits that compute the multivariate resultant.

• The paper proves that Hilbert’s Nullstellensatz over rational, number, finite, and related function fields is decidable in the counting hierarchy, improving the previous PSPACE upper bound.
• The authors construct uniform constant-depth arithmetic circuits for multivariate resultants and transfer them to CH algorithms using threshold circuits and majority quantifiers.
• The results place radical ideal membership, variety dimension, tensor-rank testing, and zero-dimensional solution counting in CH or FP^CH, while randomized reductions and open derandomization questions remain.

Overview

This paper by Andrews, Garg, and Schost improves the known complexity upper bounds for Hilbert's Nullstellensatz (HN) and several related problems in computational algebraic geometry. Given polynomials $f_1,\dots,f_m$ over a field $\bF$, HN asks whether the system $f_1=\cdots=f_m=0$ has a solution over $\overline{\bF}$. The previous best upper bound was $\PSPACE$ for the decision problem and $\FPSPACE$ for the counting version. The paper establishes that HN lies in the counting hierarchy $\CH$, and that the number of solutions of a zero-dimensional system can be computed in $\FP^{\CH}$, for coefficient fields including $\bQ$, number fields, finite fields, and rational function fields over these (2602.17904).

The counting hierarchy is defined as $\CH = \bigcup_k \C_k\P$ with $\bF$0, $\bF$1, and $\bF$2; equivalently, it captures polynomial-time computation with a constant number of polynomially-bounded majority quantifiers. It satisfies $\bF$3, so the new bounds are a genuine interpolation between the $\bF$4-hardness lower bound (conditional on standard assumptions via Koiran's $\bF$5 algorithm) and the prior $\bF$6 upper bound.

Main results

The central theorem states that HN over any of the fields $\bF$7, $\bF$8, number fields, their function-field extensions, or finite fields is decidable in $\bF$9. Three corollaries extend this reach:

• Radical ideal membership: deciding whether $f_1=\cdots=f_m=0$0 over $f_1=\cdots=f_m=0$1 is in $f_1=\cdots=f_m=0$2, via the Rabinowitsch trick. This contrasts with general ideal membership, which is $f_1=\cdots=f_m=0$3-complete (Mayr–Meyer), so a comparable improvement there would collapse $f_1=\cdots=f_m=0$4.
• Dimension: computing $f_1=\cdots=f_m=0$5 lies in $f_1=\cdots=f_m=0$6, using random hyperplane slices reduced to satisfiability queries.
• Counting solutions: the cardinality of a zero-dimensional variety is computable in $f_1=\cdots=f_m=0$7. Since this problem is complete for the class $f_1=\cdots=f_m=0$8 of Bürgisser–Cucker, the result improves $f_1=\cdots=f_m=0$9 to $\overline{\bF}$0.
• Tensor rank: deciding whether a given tensor has rank $\overline{\bF}$1 over $\overline{\bF}$2 is in $\overline{\bF}$3, by a direct reduction to HN.

The authors are explicit that the $\overline{\bF}$4-hardness of counting points explains why their techniques cannot place HN in the polynomial-time hierarchy short of an unexpected class collapse.

Technical contribution: constant-depth circuits for the resultant

The core contribution is a polylogtime-uniform family of constant-free arithmetic circuits of constant depth and size $\overline{\bF}$5 computing the multivariate resultant $\overline{\bF}$6, where $\overline{\bF}$7. The resultant vanishes exactly when a square homogeneous system has a projective solution, so efficient resultant computation yields efficient satisfiability testing after randomized reductions to square systems.

The construction combines several ingredients:

• Poisson formula recursion. The resultant factors as a product involving the smaller resultant of dehomogenized polynomials and evaluations of one polynomial at the common roots of the others. Unrolling this recursion expresses $\overline{\bF}$8 as a product over roots of intermediate systems.
• Homotopy continuation with an explicit implicit function theorem. Direct power-series descriptions of generic roots fail (the discriminant is not a square), and Newton iteration would only give polylogarithmic depth. Instead, the authors introduce a homotopy $\overline{\bF}$9 toward a carefully chosen initial system whose solutions and Jacobians are explicit (products of linear forms at Vandermonde-type points). An explicit implicit function theorem of Aizenberg–Yuzhakov then gives closed-form coefficients of the power series root trajectories $\PSPACE$0 up to degree $\PSPACE$1, computable in constant depth because they are finite sums of coefficients of explicitly bounded products. Evaluating the last polynomial at these truncated roots and interpolating in $\PSPACE$2 recovers the resultant exactly, since its degree in $\PSPACE$3 is at most $\PSPACE$4.
• Uniform interpolation machinery. Recovering coefficients from evaluations requires polylogtime-uniform constant-depth circuits for Vandermonde inversion (via Stirling numbers) and univariate/multivariate interpolation, developed carefully in the paper's uniformity framework based on direct connection languages.

A notable structural point is that the circuits are constant-free over $\PSPACE$5; resultants over other rings are obtained by lifting inputs to $\PSPACE$6 and reducing modulo $\PSPACE$7 (or modulo the ideal defining a finite field extension), exploiting that the multivariate resultant over any field is the image of the integer resultant.

Transfer from arithmetic circuits to the counting hierarchy

A transfer lemma converts polylogtime-uniform families of weakly division-free, constant-free arithmetic circuits of exponential size into $\PSPACE$8 algorithms. Each arithmetic gate becomes a $\PSPACE$9 threshold circuit implementing iterated addition, multiplication, and division on rationals represented by integer pairs, with height growth bounded by $\FPSPACE$0 per layer. Uniform exponential-size threshold circuits of constant depth are then evaluated in $\FPSPACE$1 via nested majority quantifiers, one quantifier per circuit layer. Composition lemmas handle the case where exponentially many output bits feed into polynomial-size post-processing circuits (e.g., division, modular reduction, pseudodivision by a monic lift). These ideas echo prior observations of Allender and Allen–Koiran–Reif–Ravikumar, but the authors note no prior reference derives $\FPSPACE$2 membership directly from uniform low-depth arithmetic circuits, which motivates the full treatment.

For the counting version, two further reductions are needed: a resultant-based GCD formula (from the filtering technique of the factorization literature) and a Hasse-derivative-based count of distinct roots of a univariate polynomial, both implementable in polylogtime-uniform $\FPSPACE$3 and hence composable into $\FPSPACE$4. The generalized characteristic polynomial of Canny, as adapted by Ierardi, reduces affine-system counting to counting distinct roots of a perturbed resultant.

Limitations and open questions

The paper concedes several caveats. First, the reductions from affine (non-square) systems to resultants are randomized Monte Carlo procedures with success probability at least $\FPSPACE$5; derandomization would require eliminating the random linear combinations and coordinate changes, which the authors do not address. Second, over fields of positive characteristic, Koiran-style conditional improvements remain out of reach—no $\FPSPACE$6 analogue is known for HN over finite fields, and the paper does not close this gap beyond the unconditional $\FPSPACE$7 upper bound. Third, the techniques fundamentally cannot push below $\FPSPACE$8-type hardness for the counting variant, so placing HN in a class strictly contained in the second level of $\FPSPACE$9 (e.g., $\CH$0) would need genuinely new ideas; whether $\CH$1 remains open. Fourth, the extension-field construction used when the coefficient subfield is too small is acknowledged to be far from optimal. Finally, the sparse and straight-line-program input models are handled only via dense-reduction with extension variables, preserving satisfiability but not necessarily other quantities; sharper results under succinct encodings are left unexplored.

Conclusion

This paper replaces the long-standing $\CH$2/$\CH$3 upper bounds for Hilbert's Nullstellensatz, radical ideal membership, variety dimension, solution counting, and tensor rank over algebraic closures with $\CH$4 and $\CH$5 bounds. Its technical heart—a uniform constant-depth arithmetic circuit family for the multivariate resultant built from homotopy continuation and an explicit implicit function theorem—together with a clean transfer principle to the counting hierarchy provides a template likely applicable to further problems in effective algebraic geometry whose current bounds rest on exponentially large structured linear algebra.