The SagbiHomotopy.jl package for solving polynomial systems

Barbara Betti; Viktoriia Borovik

REVIEW 3 major objections 4 minor 1 cited by

The SagbiHomotopy.jl package for solving polynomial systems

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A new Julia package, SagbiHomotopy.jl, solves horizontally parameterized polynomial systems by tracking the minimal number of homotopy paths, with a proof of optimality when the parameterization degree equals its monomial-limit degree.

desk verdict A useful, honest software paper whose optimality claim overreaches: zero-dimensional base-locus solutions can be missed even when deg(phi)=deg(phi0). read the letter →

arxiv 2506.06264 v1 pith:AHJT64ZU submitted 2025-06-06 math.AG

Barbara Betti , Viktoriia Borovik This is my paper

classification math.AG MSC 14M2513P1065H20

keywords SAGBIbasesKhovanskiitoricdegenerationhomotopycontinuationpolyhedralGröbnerhorizontallyparameterizedsystemsJuliapackage

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

The paper introduces a Julia package, SagbiHomotopy.jl, that solves systems of polynomial equations by homotopy continuation. For square horizontally parameterized systems — where each equation is a linear combination of a fixed set of polynomials that form a SAGBI basis — the package tracks exactly $\deg(\varphi)\cdot\deg(X)$ solution paths, where $\varphi$ is the rational parameterization of the underlying variety $X$. When the degree of $\varphi$ equals the degree of its monomial limit $\varphi_0$, this is the minimal possible number of paths, so the SAGBI homotopy is optimal and beats polyhedral homotopies, the default in most software. The paper proves this optimality (Proposition 3.5 and Corollary 3.6), implements the algorithm, and demonstrates it on Grassmannian, nonlinear-resonator, and coupled-cluster systems. The catch is that the equality $\deg(\varphi)=\deg(\varphi_0)$ is verified in examples but not proved in general; if it fails, the method misses solutions.

What carries the argument

A SAGBI basis — a finite set of polynomials whose leading monomials generate the initial algebra of the subalgebra they span — gives the variety a Gröbner toric degeneration to a toric special fibre. The algorithm uses the associated weight vector $\omega$ to homogenize each basis polynomial $b_{r,j}$ into $(b_{r,j})^\omega_t$, producing a flat family $S^\omega_t$ whose fibre at $t=0$ is a monomial subalgebra and whose fibre at $t=1$ is the original coordinate ring. At $t=0$ the system is a semimixed sparse system on the toric variety, solvable optimally by polyhedral homotopy via the BKK theorem, and the parameter continuation theorem guarantees that the solution paths from $t=0$ reach $t=1$. The equality $\deg(\varphi)=\deg(\varphi_0)$ is what makes the number of paths equal to $\deg(\varphi)\cdot\deg(X)$, independent of $t$.

What would settle it

Take a SAGBI basis parameterization of a multiprojective variety where $\deg(\varphi) \neq \deg(\varphi_0)$ but the base locus is empty; solve a general square horizontally parameterized system with the package and check that the output misses exactly $\deg(X)(\deg(\varphi)-\deg(\varphi_0))$ isolated solutions. If no such example exists, prove the equality $\deg(\varphi)=\deg(\varphi_0)$ for homogeneous Gröbner parameterizations, which would turn the conjecture into a theorem.

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Extended reading notes

Core claim

The central claim is that SAGBI homotopies provide an optimal start system for general square horizontally parameterized systems of the form (2), in which each equation is a linear combination of polynomials from one of the sets $B_r$ that parameterize a multiprojective variety $X$. The paper proves that if the degree of the rational map $\varphi$ equals the degree of the monomial map $\varphi_0$, then the SAGBI homotopy system has the same number of solutions outside the base locus for every $t \in \mathbb{C}$, namely $\deg(\varphi)\cdot\deg(X) = \deg(\varphi_0)\cdot\deg(X_0)$ (Proposition 3.5), and therefore the homotopy is optimal (Corollary 3.6). The algorithm replaces each basis polynomial by its weight-homogenization, solves the resulting sparse start system at $t=0$ with a polyhedral homotopy, and tracks the solution paths to $t=1$. When $\deg(\varphi) > \deg(\varphi_0)$, the method misses exactly $\deg(X)(\deg(\varphi)-\deg(\varphi_0))$ isolated solutions, as Examples 3.7 and 3.8 show.

Load-bearing premise

The optimality claim rests on the unproven equality $\deg(\varphi) = \deg(\varphi_0)$ between the degree of the rational parameterization and the degree of its monomial limit; the paper verifies it in examples and conjectures it for homogeneous Gröbner parameterizations, but does not prove it in general.

Editorial extensions

If this is right

For linear equations on a Grassmannian $\mathrm{Gr}(k,m)$, the SAGBI homotopy tracks exactly $\deg\,\mathrm{Gr}(k,m)$ paths, e.g., 14 for $\mathrm{Gr}(2,6)$, 42 for $\mathrm{Gr}(3,6)$, and 462 for $\mathrm{Gr}(3,7)$, matching the number of solutions.
For harmonic-balance equations of $N$ coupled nonlinear resonators with $M$ leading frequencies, it tracks $5^5 = 3125$ paths instead of the 59049 tracked by the default polyhedral homotopy, reducing the runtime from about seven minutes to about six seconds.
For coupled-cluster equations in quantum chemistry, the method tracks the CC degree exactly, e.g., 2883 paths for $(d,n)=(3,7)$ versus over 11000 seconds for the standard solver.
If $\deg(\varphi) = \deg(\varphi_0)$, the SAGBI homotopy is optimal for all general square horizontally parameterized systems of the form (2); if the equality fails, the algorithm warns the user and can optionally also compute the zero-dimensional base-locus solutions.

Reading between the lines

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

If the paper's conjecture that $\deg(\varphi)=\deg(\varphi_0)$ holds for all homogeneous Gröbner parameterizations is true, the method would be optimal for a wide class of Schubert and flag-variety problems beyond the Grassmannians treated here.
The one-step optimization in Algorithm 2 suggests that the SAGBI weight and the polyhedral mixed subdivision could be combined with other deformation families, potentially yielding optimal homotopies for systems that are not horizontally parameterized.
The SAGBI weight detection step is the main bottleneck for large Grassmannians (e.g., $\mathrm{Gr}(3,7)$); a more efficient detection or a direct construction of the weight from Plücker relations would extend the package's reach to $\mathrm{Gr}(4,8)$ and larger.
A natural stress test is to search for SAGBI basis parameterizations where $\deg(\varphi) \neq \deg(\varphi_0)$ but the base locus is empty; such examples would isolate the degree-drop failure from the base-locus failure and clarify which systems the package could silently solve incorrectly.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a real software paper with a genuinely useful package, but the advertised optimality claim overreaches. The core algorithm is not new — the authors say so themselves: theoretically they use the same algorithm as BSW20. What is new is the Julia implementation, the multiprojective extension, the one-step homotopy in Section 3.2, and the integration with SAGBI detection. Those are legitimate additions. The path-count reductions are impressive: 3,125 tracked paths versus 59,049 for the resonator system, and 42 paths for Gr(3,6). The code and examples are on MathRepo, so the numbers can be checked. Credit where due: the paper is honest about what is inherited, and the package seems to work for what it tracks.

Now the soft spots. Proposition 3.5 is the load-bearing result, but its proof is a sketch: it appeals to flat degeneration and the generalized BKK theorem, and the key assumption deg(phi)=deg(phi0) is only verified case by case and conjectured in general. If that equality fails, the algorithm provably misses deg(X)*(deg(phi)-deg(phi0)) solutions, as Examples 3.7 and 3.8 show. That is disclosed, and it is not a fatal flaw.

More serious is a gap between Proposition 3.5 and Corollary 3.6. The proposition counts solutions outside the base locus; the corollary drops the qualifier. The stress test is correct: take B={x,y,xy} in C[x,y] with omega=(2,1). This is a SAGBI basis, deg(phi)=deg(phi0)=1, and the degree check in Algorithm 1 passes. The base locus is the origin. A general system a x+b y+c xy=0, d x+e y+f xy=0 has exactly two isolated solutions: the origin and one torus point. The SAGBI homotopy tracks one path and, by default, returns only the torus point. So Corollary 3.6 is false as stated. The authors know base-locus solutions can be missed (Example 4.3), and they even note that getBaseLocus=true can miss the origin because the polyhedral start system lives in the torus. But the corollary and abstract still promise all solutions. A revised version should state optimality only outside the base locus and document that zero-dimensional base-locus components contribute additional isolated solutions that the default pipeline misses.

Minor point: the timing comparisons are not fully apples-to-apples; SAGBI detection can dominate the total time, and for the smallest cases plain solve is faster. The large cases still make the case.

Overall: the package is useful, the write-up is clear, and the flaw is fixable. This deserves a serious referee. I would send it to review, expecting major revision to correct the optimality statement and to tighten the proof of Proposition 3.5.

Referee Report

3 major / 4 minor

Summary. The paper presents SagbiHomotopy.jl, a Julia package that implements SAGBI homotopy continuation for square horizontally parameterized polynomial systems, i.e., systems in which each equation is a linear combination of a fixed set of polynomials coming from a (multi)projective parameterization. The main theoretical claim is Proposition 3.5: if the degree of the parameterization phi equals the degree of the monomial map phi0 of the special fiber, then the SAGBI homotopy system has the same number of solutions outside the base locus for every t, namely deg(phi)*deg(X). Corollary 3.6 then asserts optimality of the SAGBI homotopy for general square horizontally parameterized systems. The package detects a SAGBI weight, solves a monomial start system by polyhedral homotopy, and tracks paths to the target system. Applications to linear sections of Grassmannians, harmonic balance systems from nonlinear dynamics, and coupled cluster equations from quantum chemistry are reported with computational timings.

Significance. The implementation is a concrete, reproducible artifact with public code and MathRepo examples, and the reported timings show potentially large speedups over the default polyhedral homotopy in HomotopyContinuation.jl. The extension of Khovanskii/SAGBI homotopies from the single projective setting to multiprojective varieties and the avoidance of computing the defining ideal are useful contributions. If the optimality statement is properly qualified, the package provides a practical method for a structured family of systems arising in several applications. The paper's strength is that it ships reproducible code, named algorithms, and explicit computational comparisons; its main weakness is that the central optimality claim is stated more broadly than the theory supports.

major comments (3)

[Section 3.1, Corollary 3.6 and Algorithm 1] Corollary 3.6 is false as stated because the equality deg(phi)=deg(phi0) does not exclude zero-dimensional base-locus solutions, which are isolated solutions of the target system but are not tracked by the SAGBI homotopy. Take n=2, m=1, and B={x,y,xy}. With weight omega=(2,1), B is a SAGBI basis, and both phi and phi0 are the map [x:y:xy], so deg(phi)=deg(phi0)=1 and deg(X)=1. The base locus is the single point (0,0). A general system f1=a x+b y+c xy, f2=d x+e y+f xy has exactly two isolated solutions: the origin and one solution in (C*)^2. Algorithm 1 starts from the identical monomial system F0=F, and the polyhedral homotopy tracks exactly one torus solution, silently returning only the nonzero solution. Thus the advertised path count deg(phi)*deg(X)=1 is not the number of isolated solutions of the target system, and Algorithm 1's stated output is incomplete. The qualification 'outside base locus' in Proposition 3.5 must be carried into Corollary 3.6, into Algorithm 1's output specification, and into the abstract; zero-dimensional base-locus points require a separate tracking mechanism, and the current optional getBaseLocus does not recover the origin, as Example 4.3 itself shows.
[Section 3.1, Proposition 3.5] The proof of Proposition 3.5, which is the main mathematical justification for optimality, is only sketched by a reference to flat degeneration and to the generalized BKK theorem from [Bor+24]. The manuscript should either supply a complete proof or state precisely which theorem in [Bor+24] implies the claimed equality of counts outside the base locus for every t, including the correspondence between solutions of F_t^omega and points of the linear section X_t cap L. This is load-bearing because Corollary 3.6 and the optimality claims in the abstract and introduction rest on it.
[Section 3.1 and Section 5] The hypothesis deg(phi)=deg(phi0) is not proved for any of the application families in Section 5; it is verified example by example and conjectured only for homogeneous Groebner parameterizations. Since Example 3.8 shows that when the hypothesis fails the algorithm misses deg(X)*(deg(phi)-deg(phi0)) isolated solutions, the optimality claims should be presented as conditional on a verified equality. The runtime degree check in sagbi_homotopy is a computational guard, not a proof, and the paper should say so explicitly, or prove the equality for the Grassmannian and harmonic-balance families if such a proof exists.

minor comments (4)

[Example 4.4] The code fragment '@var z[1:z]' appears to be a typo; it should define a vector of the correct length, e.g., '@var z[1:15]' for Gr(2,6).
[Table 2] There are typographical artifacts in the timing columns, such as '3.94 4 .21s' and '2.49 1 .902s'; these should be cleaned up.
[Algorithm 1, Step 3] Step 3 should state explicitly that the polyhedral homotopy solves F0 in the torus (C*)^n; this is the reason why isolated base-locus solutions with zero coordinates are not found by default.
[Example 4.3] The output message in Example 4.3 is confusing: it prints an error saying the SAGBI homotopy will not find all solutions and then immediately prints 'SAGBI homotopy successfully completed with 2 solutions'. Consider rewording the success message to indicate that the computation is incomplete rather than successful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: path counts derive from independent degree invariants, and all cited prior work is published with independent proofs.

full rationale

The central derivation (Proposition 3.5 and Corollary 3.6) counts solutions of the SAGBI homotopy family as deg(phi)*deg(X) = deg(phi0)*deg(X0). This count is not fitted to the number of solutions observed in the examples; it is computed from the independent invariants deg(phi) (the degree of the rational parameterization) and deg(X) (the degree or multidegree of the variety), with deg(X) = deg(X0) following from flatness of the Groebner degeneration. The condition deg(phi) = deg(phi0) is explicitly stated as an assumption, verified case-by-case, and conjectured for homogeneous Groebner parameterizations; it is not manufactured by the homotopy. The cited prior works ([BDS24], [Bor+24], [BSS25], [BB25]) provide published algorithms and theorems with independent proofs and implementations, so even where the author sets overlap, they constitute real evidence rather than a circular self-citation chain. The paper itself acknowledges the zero-dimensional base-locus limitation in Example 3.8 and the following remark, noting that such solutions are missed unless getBaseLocus is used; this is a correctness caveat about the wording of Corollary 3.6, not a circular reduction of the path count to the homotopy's own output. No circular step is present.

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

The paper introduces no fitted constants or new mathematical entities. The method rests on standard toric degeneration facts, BKK/Kushnirenko degree computations, and the Morgan-Sommese continuation theorem. The only load-bearing condition specific to this work is the degree equality deg(phi)=deg(phi0), which is verified in examples but not proven in general; the paper conjectures it for homogeneous Groebner parameterizations. SAGBI detection is outsourced to BDS24.

assumptions (5)

standard math Flatness of Groebner degeneration: C[z,t]/I^v_t is flat as a C[t]-module (Eisenbud, Thm 15.17).
Invoked in Proposition 2.6 to produce a flat family whose general fiber is C[z]/I and special fiber is C[z]/in_v(I); flatness is what allows transferring degrees between fibers in Lemma 2.2/2.3 and Proposition 3.5.
standard math Parameter continuation theorem of Morgan and Sommese [MS89].
Used in Section 1 to justify that solution paths exist for the homotopy H and that tracking yields all isolated solutions when the start system has the right number of zeros.
standard math Kushnirenko and BKK theorems compute degrees of toric varieties as (mixed) volumes.
Theorems 2.4 and 2.5 give deg(X0) and the mixed volumes used to determine the number of solutions of the sparse start system F0.
ad hoc to paper The user-supplied parameterization satisfies deg(phi) = deg(phi0).
Corollary 3.6 and Proposition 3.5 assume this equality; the paper verifies it per example and conjectures it for homogeneous Groebner parameterizations but gives no proof. When it fails, the algorithm misses solutions (Example 3.8).
domain assumption SAGBI detection algorithm from BDS24 terminates and returns a valid weight if one exists.
Algorithm 1 Step 1 and Algorithm 2 Step 1 rely on this external algorithm; the paper modifies it to stop at the first valid weight (Section 4 remark).

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Cite this review

Pith. "Pith review of The SagbiHomotopy.jl package for solving polynomial systems." pith.science (2026). https://pith.science/paper/AHJT64ZU

@misc{pith2026250606264,
  author       = {Pith},
  title        = {Pith review of: The SagbiHomotopy.jl package for solving polynomial systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHJT64ZU}},
  note         = {Machine review of arXiv:2506.06264}
}

read the original abstract

We present the Julia package SagbiHomotopy.jl for solving systems of polynomial equations using numerical homotopy continuation. The package introduces an optimal choice of a start system based on SAGBI homotopies. For square horizontally parameterized systems, where each equation is a linear combination of a given set of polynomials, SAGBI homotopies significantly reduce the number of solution paths to track compared to polyhedral homotopies currently used by default in most software for numerical homotopy continuation. We illustrate our framework with a variety of examples, including problems arising in chemistry and physics.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Degenerating Discriminants
math.AG 2026-07 conditional novelty 6.0 of 10

Under a Gröbner degeneration, the limiting dual hypersurface decomposes into duals of Whitney strata of the special fiber with multiplicities given by Sabbah's formula.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Bertini: Software for Numerical Algebraic Geometry

[Bat+13] Daniel J. Bates et al. “Bertini: Software for Numerical Algebraic Geometry”. In:Available at bertini.nd.edu(2013). [BB25] Viktoriia Borovik and Paul Breiding. “A short proof for the parameter continuation theorem”. In:Journal of Symbolic Computation127 (2025), p. 102373. [BDS24] Viktoriia Borovik, Timothy Duff, and Elima Shehu.SAGBI and Gröbner b...

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Homotopy continuation methods for coupled-cluster theory in quantum chemistry

[Fal11] GregoryFalkovich. Fluid Mechanics. Vol. 9781107005. Cambridge: Cambridge University Press, 2011, pp. 1–167. [FG22] Claudia Fevola and Christiane Görgen. “The mathematical research-data repository MathRepo”. In:Computeralgebra-Rundbrief 70 (2022). [FL23] Fabian Faulstich and Andre Laestadius. “Homotopy continuation methods for coupled- cluster theo...

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[MS89] Alexander P. Morgan and Andrew J. Sommese. “Coefficient-parameter polynomial con- tinuation”. In:Applied Mathematics and Computation29.2 (1989), pp. 123–160. [OW24] Nida Kazi Obatake and Elise Walker. “Newton-Okounkov bodies of chemical reaction systems”. In:Advances in Applied Mathematics155 (2024), p. 102672. [RS90] Lorenzo Robbiano and Moss Swee...

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