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Solving bihomogeneous polynomial systems with a zero-dimensional projection

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Pith's one-line read The paper establishes multiplication matrices for bihomogeneous systems with finite projection and proves their eigenvalues recover the projected points.

desk verdict A novel and mostly sound generalization of FGLM to positive-dimensional bihomogeneous ideals with finite projection, but the general admissible-bidegree bound in Theorem 4.2 is false as stated and needs a simple fix. read the letter →

arxiv 2502.07048 v2 pith:BFC64MBO submitted 2025-02-10 math.AC

Matías Bender , Laurent Busé , Carles Checa , Elias Tsigaridas This is my paper · ORCID
classification math.AC MSC 13P1014Q2013D0268W30
keywords bihomogeneouspolynomialsystemszero-dimensionalprojectionmultiplicationmatricesFGLMalgorithmeliminationidealadmissiblebidegreeeigenvaluemethodssingleexponentialcomplexity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a way to solve bihomogeneous polynomial systems—systems in two groups of variables, separately homogeneous, whose solution variety sits in a product of two projective spaces—when the projection onto the first group of variables is a finite set of points. The authors introduce linear multiplication maps that act on a finite-dimensional piece of the quotient ring at a carefully chosen 'admissible' bidegree, without first computing an elimination ideal. They prove that these maps encode the localized elimination ideal of the projection, so a generalized FGLM algorithm recovers a Gröbner basis for it, and that the eigenvalues of the maps are exactly the projected points, with multiplicities given by lengths of local modules. If correct, this gives a single-exponential algebraic and numerical pipeline for a class of systems that covers parameterized eigenvalue problems and many elimination problems.

What carries the argument

The central object is an admissible bidegree $(a,b)$ together with an admissible linear form $h$ of bidegree $(1,0)$: a degree at which the Hilbert function of $R/I$ is stable in the $x$-degree and the quotient $R/(I,h)$ vanishes at $(a,b)$. At such a bidegree, multiplication by $h$ is invertible on the finite-dimensional space $(R/I)_{a,b}$, so every rational function $g/h^k$ with $\deg g = k$ gives a well-defined linear map. These maps form a commutative ring whose kernel is the localization of the annihilator module $J_b = (I : m_y^b) \cap R_X$, and that kernel statement is the mechanism carrying the whole argument: it turns the projected variety into the spectrum of a finite-dimensional algebra of matrices.

What would settle it

Run the constructive algorithm on a random bihomogeneous ideal with finite projection and positive-dimensional fibres at the bidegree given by Theorem 4.2: if the Hilbert-function equalities $\mathrm{HF}_{R/I}(a,b)=\mathrm{HF}_{R/I}(a+1,b)$ and $\mathrm{HF}_{R/(I,h)}(a,b)=0$ fail for every $h$, then the claimed existence bound is false. Alternatively, in a case where the full variety is zero-dimensional, check that the multihomogeneous Macaulay bound of Theorem 4.1 is admissible; the authors state no counterexample is known for positive-dimensional fibres, so a single such counterexample would refute the conjectured strengthening.

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Extended reading notes

Core claim

Fixing a bihomogeneous ideal $I$ and an admissible bidegree $(a,b)$ satisfying $\mathrm{HF}_{R/I}(a,b)=\mathrm{HF}_{R/I}(a+1,b)$ and $\mathrm{HF}_{R/(I,h)}(a,b)=0$ for a degree-$(1,0)$ form $h$, the paper constructs maps $m_{g/h^k} = \bar m_{h^k}^{-1} \circ \bar m_g$ on $(R/I)_{a,b}$. Theorem 3.6 states that the kernel of the induced ring homomorphism $(R_{X,h})_0 \to k[m_{g'}]$ is exactly $(J_b \otimes_{R_X} R_{X,h})_0$, where $J_b = (I : m_y^b) \cap R_X$ is the annihilator of $(R/I)_{\ast,b}$. Consequently FGLM on these matrices yields a Gröbner basis of the localized elimination ideal, and Theorem 3.13 identifies the exponents of the characteristic polynomial with the lengths of the local modules $(R/I)_{\ast,b} \otimes_{R_X} (R_X)_{p_\xi}$. Thus the eigenvalues compute the zero-dimensional projection, possibly with extra points that disappear as $b$ grows.

Load-bearing premise

The whole construction rests on the existence of a bidegree $(a,b)$ and a linear form $h$ satisfying the two Hilbert-function equalities of Definition 2.1, with $h$ a non-zero-divisor on the saturated module $R_X/(J_b : m_x^\infty)$; the paper proves such degrees exist only under the finite-projection hypothesis together with a loose bound, while its sharper multihomogeneous Macaulay bound is proved only when the whole variety $V(I)$ is zero-dimensional, and small-characteristic fields require extra care.

Editorial extensions

If this is right

  • A Gröbner basis of the elimination ideal of the projection can be computed by running FGLM on the new multiplication matrices, so no elimination ordering or prior elimination step is required.
  • The eigenvalues of any multiplication map are the values of that rational function on the points of $\pi_b(V(I))$; evaluating several maps gives numerical approximations of the projected points, and the characteristic polynomial's multiplicities equal the lengths of the local modules $M_{b,\xi}$.
  • Increasing the second degree $b$ shrinks the computed set toward the true projection, and Proposition 4.6 gives a symbolic or numerical test to discard points that are not in $\pi(V(I))$.
  • Under the finite-projection hypothesis, the complexity of recovering a Gröbner basis for $(J_b \otimes R_{X,x_0})_0$ is single exponential in the input size, $O\!\left(l^4 (n+m+nm) A^{4n} B^{4(m+mn)}\right)$, where $A,B$ are the maximum $x$- and $y$-degrees of the generators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The construction suggests a degroebnerization-style solver for parameterized systems: build the matrices directly from truncated Macaulay matrices, extract eigenvalues numerically, and only certify with Gröbner methods; this could bypass expensive elimination orders for moderate parameter counts.
  • If the multihomogeneous Macaulay bound holds for positive-dimensional fibres—the paper leaves this open—the loose bound of Theorem 4.2 would tighten to a practically useful regime; conversely a counterexample would delineate exactly where the method's complexity guarantee stops.
  • The link to bigeneric initial ideals indicates that in generic coordinates the admissible degrees might be read directly off the leading monomials of a Gröbner basis, potentially giving a pre-processing test for admissibility without rank computations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies bihomogeneous ideals I ⊂ k[x0,...,xn,y0,...,ym] defining biprojective varieties V(I) whose projection onto P^n is a finite set of points. It introduces 'admissible bidegrees' (a,b) for which the Hilbert function of R/I stabilizes in the x-degree and R/(I,h) vanishes at degree (a,b) for some linear form h. At an admissible bidegree, the paper defines generalized multiplication maps m_{f/h^k} on the finite-dimensional vector space (R/I)_{a,b}, shows they form a commutative ring isomorphic to the localization of the annihilator ideal J_b (Theorem 3.6), recovers a Gröbner basis of the localized elimination ideal via FGLM (Corollary 3.7), and identifies eigenvalues and their multiplicities with the projected points and lengths of local modules (Theorems 3.9 and 3.13). It then gives complexity bounds: the multihomogeneous Macaulay bound for zero-dimensional V(I) (Theorem 4.1) and a loose bound for positive-dimensional fibers via generalized Koszul complexes (Theorem 4.2), leading to a single-exponential complexity statement (Theorem 4.5). Finally, it relates admissible bidegrees to the bigeneric initial ideal.

Significance. The main construction is novel and potentially valuable: it extends multiplication-map techniques beyond zero-dimensional systems while avoiding explicit elimination. The eigenvalue characterization with multiplicities as lengths of local modules is a clean generalization of the classical zero-dimensional case. The paper includes a worked example (Examples 2.10, 3.8, 4.3) and the central theorems (3.6, 3.9, 3.13) have coherent proofs, though several steps are delegated to cited lemmas (Bayer–Stillman, spectral sequences). However, the single-exponential complexity guarantee for the positive-dimensional case rests on Theorem 4.2, which is false as stated; the error is fixable by adding the generation-degree condition. The connection to bigeneric initial ideals in Section 5 relies on the authors' own [6], but this is not circular for the main multiplication-map results.

major comments (2)
  1. [Section 4, Theorem 4.2] The stated bound is false. Let k be any infinite field, R = k[x0,x1,y0,y1,y2] (so n=1, m=2), and I = ⟨y0^100, x0 y0 − x1 y1, x1^10⟩. I is bihomogeneous with bidegrees (0,100), (1,1), (10,0), and V(I) = {[1:0]} × {[0:y1:y2]} ≅ P^1, so π(V(I)) is a single point. Theorem 4.2 asserts that every b ≥ (100+1+0) − 2 = 99 is admissible in the second coordinate; for b=99 the condition gives a ≥ (0·N_{−1} + 1·N_{98} + 10·N_{99}) − 1 = 55449. But Definition 2.1 requires I to be generated in bidegrees ≤ (a,b), and the generator y0^100 has b-degree 100 > 99, so no bidegree (a,99) can be admissible. The proof of Theorem 4.2 establishes only the Hilbert-function stabilization conditions and silently drops the generation condition. This is load-bearing because Theorem 4.5 relies on Theorem 4.2 for the unconditional complexity bound. The fix is to require b ≥ max{Σ_i b_i − m, max_i b_i}; with that amendment the rest of the argument appears to go through.
  2. [Section 4, Theorem 4.5] The proof states without derivation that the size of the multiplication matrices is at most S := l^{n+m+nm} A^n B^{m+mn}/(n!(m!)^{n+1}) for (A,B) = (max_j deg_x f_j, max_j deg_y f_j). Even after correcting Theorem 4.2, the admissible first coordinate a is of order (Σ a_i) · C(b+m,m), which can be far larger than A, and b can be as large as Σ b_i − m, which can exceed B. The claimed bound S may still be valid as a worst-case estimate, but the proof needs to supply the explicit calculation, since the current statement is not immediate from the corrected Theorem 4.2.
minor comments (6)
  1. [Notation] In the notation paragraph, 'localization of RX by he multiplicative set' should read 'the multiplicative set', and 'bihomogeheous' is a typo for 'bihomogeneous'.
  2. [Remark 2.2] The 'if and only if' statement about existence of admissible bidegrees should be qualified by the existence of an admissible linear form over the ground field; small-characteristic fields may fail after generic change of coordinates (as Section 4 later acknowledges).
  3. [Proof of Theorem 3.6] The expression 'f hayβ' should be 'f h^a y^β' (with a superscript on h); as printed it is confusing.
  4. [Corollary 5.3] The notation 'HF(I,h)(a−1,b)' is ambiguous; it should be written as HF_{R/(I,h)}(a−1,b) or defined explicitly.
  5. [Example 4.3] The claimed upper bound '(83,6)' is computed from the uncorrected Theorem 4.2; after the required amendment to the bound, this example should be updated or recomputed.
  6. [Theorem 4.1] The proof of Theorem 4.1 is only sketched with a reference to [15, Prop. 3.15]; given that the statement is used as a black box for later results, the authors should either reproduce the spectral-sequence comparison or give a more precise citation to the exact statement.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular dependence in the central construction; only a minor, non-load-bearing self-citation in Section 5.

full rationale

The main derivation chain is self-contained. Given an admissible bidegree, Section 3 constructs the generalized multiplication maps from Hilbert-function data, proves the kernel statement in Theorem 3.6 using Lemma 2.4, Corollary 2.6, and localization arguments, and then invokes the classical FGLM algorithm in Corollary 3.7. The eigenvalue and multiplicity theorems, Theorems 3.9 and 3.13, are derived in-paper from Lemma 3.1, Theorem 3.6, Lemma 3.12, and standard commutative-algebra facts, not from the paper's own conclusions. No fitted parameter is renamed as a prediction, no target quantity is used in its own definition, and no uniqueness theorem is invoked to force the construction. The only noticeable self-citation is [6], by the same four authors, used in Section 5 to relate admissible bidegrees to bigeneric initial ideals. That section is a supplementary structural comparison and is not needed for Theorems 3.6, 3.9, 3.13, or for the main complexity mechanism, so the self-citation is minor and not load-bearing. The alleged counterexample to Theorem 4.2 concerns a missing hypothesis in the admissible-bidegree bound, namely the generation-degree condition b >= max_i b_i from Definition 2.1; if it stands, it is a correctness gap in an upper-bound theorem, not a circular reduction of the paper's results to their own inputs. Accordingly, no circular step is identified, and the score reflects only the minor self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All ledger entries are mathematical assumptions inherited from the field, not fitted constants or invented entities. The paper introduces no empirical parameters. The main domain assumption is the finite-projection hypothesis; the standard-math axioms are the technical machinery used in the proofs.

assumptions (5)
  • domain assumption The base field k is infinite and, when needed, its characteristic is large enough for generic changes of coordinates.
    Used to ensure a generic linear form h is admissible and non-zero-divisor (Lemma 2.8, Corollary 2.9); small-characteristic cases are not handled.
  • domain assumption For every ideal considered, the projection π(V(I)) ⊂ P^n is a finite set of points.
    This is the defining hypothesis of the paper; Remark 2.2 ties it to existence of admissible bidegrees.
  • standard math Standard facts on Castelnuovo-Mumford regularity, local cohomology and the Grothendieck-Serre formula hold as invoked.
    Used in Theorem 2.3, Lemma 3.12 and Theorem 4.1 to pass from Hilbert function stabilization to vanishing of local cohomology.
  • standard math Bayer-Stillman criterion [5, Lemma 1.9 and Lemma 1.4] gives the stated correspondences between syzygies, colons and zero-divisors.
    Central to Theorem 2.3 and Lemma 2.8; the paper cites rather than proves these lemmas.
  • standard math Properties of generalized Koszul complexes and annihilation by the 0th Fitting ideal [38, App. C.2] hold.
    Used in the proof of the loose admissible-degree bound, Theorem 4.2.

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Pith. "Pith review of Solving bihomogeneous polynomial systems with a zero-dimensional projection." pith.science (2026). https://pith.science/paper/BFC64MBO

@misc{pith2026250207048,
  author       = {Pith},
  title        = {Pith review of: Solving bihomogeneous polynomial systems with a zero-dimensional projection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFC64MBO}},
  note         = {Machine review of arXiv:2502.07048}
}
read the original abstract

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Gr\"obner bases. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Gr\"obner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal.

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