Sign conditions for the existence of at least one positive solution of a sparse polynomial system

Alicia Dickenstein; Fr\'ed\'eric Bihan; Magal\'i Giaroli

REVIEW 2 major objections 3 minor 23 references

Sign conditions for the existence of at least one positive solution of a sparse polynomial system

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes sign conditions on the exponents and coefficients of a sparse polynomial system that guarantee the existence of at least one positive real solution, via Gale duality and degree theory.

desk verdict New sufficient sign conditions for positive roots of sparse systems, mostly well-supported but Theorem 3.8 overclaims: the proof handles only uniform C, so the nonuniform case needs either a repaired argument or a restricted statement. read the letter →

arxiv 1908.05503 v2 pith:SQDUYTH6 submitted 2019-08-15 math.AG

Frédéric Bihan , Alicia Dickenstein , Magalí Giaroli This is my paper · ORCID

classification math.AG MSC 14P1014M2552B40

keywords sparsepolynomialsystempositivesolutionGaledualitydegreetheorydominatingmatrixsignconditionsorientedmatroidlatticeideal

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 asks when a system of $d$ generalized polynomials in $d$ variables has at least one positive solution, and answers with sign conditions that can be checked from the exponents and coefficients alone. The key move is Gale duality: instead of solving the original system, one studies an equivalent system of equations on a bounded polytope $\Delta_P$ built from a Gale dual of the coefficient matrix. The main theorem (Theorem 3.8) says that if certain submatrices of a Gale dual $B$ of the exponent matrix are not weakly mixed, and if boundary sign inequalities involving $B$ and a Gale dual $D$ of $C$ hold with at least one strict product and zero compatibility, then the Gale map points inward on the whole boundary, so degree theory forces a zero, hence a positive solution of the original system. A cleaner sufficient criterion (Theorem 4.6) states that when both matrices are uniform, it is enough that $A$ admit a dominating Gale dual $B$ and that each column of $B$ have a sign pattern realizable in $\ker C$. In the integer-exponent case the dominance condition is related to complete-intersection lattice ideals, and the same machinery yields existence of real nonzero solutions.

What carries the argument

The load-bearing object is the Gale dual pair: a matrix $B\in\mathbb{R}^{n\times k}$ whose columns span $\ker A$, and a matrix $D\in\mathbb{R}^{n\times(k+1)}$ whose rows $P_i$ span the kernel of $C$. These convert the original system into the Gale dual system $\prod_i \langle P_i,y\rangle^{b_{ij}}=1$, $j=1,\dots,k$, whose solutions in $\Delta_P$ are in bijection with positive solutions of the original system (Theorem 2.5). The sign of the resulting Gale map on each face of $\Delta_P$ is controlled by Lemma 3.4, and Theorem 3.1, a Brouwer degree theorem for inward-pointing vector fields, turns the sign conditions into a forced zero. Thus the machinery replaces root finding with a combinatorial sign check on boundary indices.

What would settle it

Take the configuration and coefficient matrix of Example 3.10 but replace one column of $C$ by a positive multiple of another, breaking uniformity while keeping $0\in C^\circ$; compute a Gale dual $D$, form the reduced Gale map $\bar g$ with the constants $c_j$, and check on the face where the two proportional rows vanish whether the conclusion of Lemma 3.4 still holds. If any component of $\bar g$ has the opposite sign, or if $\bar g$ has a boundary zero while condition (2) holds, the stated generality of Theorem 3.8 fails.

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Extended reading notes

Core claim

The central claim is that the existence question for a sparse system $f_i(x)=\sum_j c_{ij}x^{a_j}=0$ can be decided by sign data on a pair of Gale dual matrices. Let $A$ be the matrix with columns $(1,a_j)$, let $B$ be a Gale dual of $A$, and let $D$ be a Gale dual of $C$ chosen so that the polytope $\Delta_P=\{y_0=1,\langle P_i,y\rangle>0\}$ is bounded. After passing to the Gale dual system, the sign of every component of the Gale map $g=(g_1,\dots,g_k)$ on each face of $\Delta_P$ is determined by the signs of the entries of $B$ and $D$. Theorem 3.8 asserts that when, for every face $L$, the reduced submatrix $\overline{B}_{\overline{L}}$ is not weakly mixed, and when for each boundary index $i$ the products $\overline{b}_{rj}d_{ij}$ are all nonnegative, with at least one positive product and zero compatibility, the map $-g$ points strictly inward along the entire boundary of $\Delta_P$. Brouwer degree then gives a zero of $g$ in $\Delta_P$, which by the Gale-duality bijection is a positive solution of the original system.

Load-bearing premise

The main theorem is stated for all coefficient matrices $C$ of maximal rank, but the proof, after merging proportional rows of the Gale dual $D$ into equivalence classes, says it is 'not difficult to see' that the boundary-sign lemma survives and then declares 'we assume from now on that $C$ is uniform'; the full-strength theorem depends on that unverified reduction.

Editorial extensions

If this is right

When $A$ and $C$ are uniform and $A$ has a dominating Gale dual $B$ whose columns realize sign patterns from $\ker C$, the system has at least one positive solution (Theorem 4.6).
In the circuit case $k=1$, the condition in Theorem 3.8 is equivalent to $n_A(C)$ being odd, so the sufficient condition can guarantee more than mere existence: it implies an odd number of positive roots (Example 3.9).
If $A$ and $C$ are $I$-compatible for some subset $I$, with uniform matrices $A,C,\overline{C}$ and $0\in C^\circ$, then $n_A(C)>0$ (Theorem 5.8).
For integer exponent sets, existence of a mixed dominating Gale dual is equivalent to the associated lattice ideal being a complete intersection; combined with the sign condition this gives positive roots (Corollary 6.2).
The same Gale-duality framework yields a count of real nonzero solutions in terms of solutions of the Gale dual system away from the hyperplane arrangement, and the count is even when $\operatorname{rk}([A]_2)\le d$ (Theorem 6.10).

Reading between the lines

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

The sign conditions in Theorem 3.8 depend only on the oriented matroids of $A$ and $C$, i.e. on the signs of maximal minors, so they are decidable by inspecting sign patterns rather than by numerical solving; the paper does not package this as an algorithm.
The degree argument carries a multiplicity consequence the paper does not state in its main theorem: under the hypotheses of Theorem 3.8, if the zero in $\Delta_P$ is nondegenerate and its Jacobian has sign $(-1)^{k+1}$, then there are at least three zeros, hence at least three positive solutions, by the last statement of Theorem 3.1.
The $I$-compatibility criterion (Theorem 5.8) suggests a recursive verification procedure: check the subconfigurations indexed by $I$, then test only that the remaining points lie in the convex hulls of $(d+1)$-subsets of $I$.
In the integer setting, the same boundary-sign machinery could be applied cone by cone to the real-torus formulation of Section 6, effectively giving a 'real version' of Theorem 3.8; the paper states the counting result but does not formulate such a separate real theorem.

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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

Short version: the paper does something new and mostly does it well. It gives the first general sign conditions on the support and coefficients of a sparse system that guarantee at least one positive root, using Gale duality together with degree theory. The main engine is Theorem 3.8, and the later sections translate its hypotheses into more checkable conditions. I think this deserves peer review, but the statement of Theorem 3.8 is wider than its proof.

What is actually new: Theorem 3.8 and the corollaries that follow from it (Theorems 4.6, 5.8, 6.10) are not in the existing literature. The circuit case k=1 recovers the known Descartes-type parity condition, but for general configurations the sign conditions are new. The authors are also honest that their conditions are sufficient and not necessary: Example 3.9 makes this explicit, and Example 3.10 shows the condition does not imply uniqueness. That honesty is worth respecting.

Where the paper is solid: the central machinery is mostly coherent. Lemma 4.2 is correct, and Theorem 4.6 follows from it together with Lemma 4.5, given the mixed-dominating matrix framework from Fischer-Shapiro. The Singular computation in Example 3.10 is real evidence and it is presented in a responsible way: for c=1/2 the system has 3 positive roots and for c=8/7 it has 1, which is exactly the behavior you would expect if the sufficient condition is not forcing uniqueness. No circularity problem: the new theorems build on earlier Gale-duality and degree-theory tools, but they are not derived from the results they cite as prior bounds.

Soft spots: the first and main one is in the proof of Theorem 3.8. After reducing proportional rows of D into equivalence classes and defining the reduced map, the text says it is 'not difficult to see' that Lemma 3.4 and Corollary 3.5 extend, and then immediately says 'we assume from now on that C is uniform.' But Theorem 3.8 is stated for all full-rank C with 0 in C^deg and bounded full-dimensional Delta_P; nonuniform C are explicitly included. The inward-vector argument as written works for the uniform case only. I think the extension is probably repairable, because the positive constants c_j do not change signs, but that repair is not in the manuscript. The practical damage is limited because most applications afterward (Theorems 4.6, 5.8, 6.10) assume uniformity anyway, but the theorem as stated overclaims.

Second minor issue: Propositions 4.3 and 4.4 are asserted to extend from integer to real matrices because the proofs in the cited papers 'clearly' carry over. I did not find a counterexample and I believe it is true, but it is an unverified assertion in the text.

Who is this for: people working in fewnomial theory, Gale duality for sparse systems, or positive steady states in chemical reaction networks. It is a serious contribution with a fixable gap, and it deserves a serious referee rather than a desk rejection. My recommendation: send it to peer review, with the referee asked to focus on the nonuniform step in Theorem 3.8; the authors should either prove the extension or restrict the theorem to uniform C.

Referee Report

2 major / 3 minor

Summary. The paper studies sparse generalized polynomial systems (1.1) with exponent set A and coefficient matrix C, and gives sufficient conditions, based on Gale duality and degree theory, for the existence of at least one positive solution. The authors construct the polytope ΔP from a Gale dual of C, transform the original system into a Gale dual system on ΔP, and apply a Brouwer degree argument to an inward-pointing vector field. The main result, Theorem 3.8, states sign conditions on a Gale dual B of the exponent matrix and on the Gale configuration of C; Sections 4 and 5 provide more checkable sufficient conditions using mixed dominating matrices and geometric compatibility, and Section 6 relates these conditions to toric ideals and extends the search to real torus solutions.

Significance. If correct, the paper offers a new and fairly general mechanism for guaranteeing positive solutions of sparse polynomial systems, complementing injectivity- and Descartes-type results. The framework elegantly combines oriented matroid sign conditions with degree theory, and the connection to complete intersection lattice ideals in the integer case is natural and likely useful. The paper contains concrete worked examples verified with Singular, which is a strength. However, the main theorem as stated is broader than what the proof actually establishes, and this gap must be addressed before the result can be fully relied upon.

major comments (2)

[Proof of Theorem 3.8, after Eq. (3.7)] The proof does not handle nonuniform coefficient matrices C, although Theorem 3.8 is stated with that generality. Immediately after defining the reduced matrix \bar B and the equivalent system (3.5)-(3.7), the text asserts without proof that 'the conclusions of Lemma 3.4 and Corollary 3.5 hold for the map \bar g' and then says 'To simplify the notation, we assume from now on that C is uniform.' Lemma 3.4 and Corollary 3.5 are proven only under the assumption that C is uniform, and the proof of Lemma 3.4 explicitly uses uniformity to ensure that no two linear forms p_i are proportional. Since Theorem 3.8 only assumes that C has maximal rank, that 0∈C^∘, and that ΔP is full-dimensional and bounded, nonuniform C are included in the statement. The inward-vector argument on all faces of ΔP then relies on a sign computation that has not been established for the reduced map \bar g with the positive constants c_j. This is a load-bearing gap: the theorem's full generality rests on an unproved extension. The authors should either prove the extension for the reduced map explicitly or restrict the statement of Theorem 3.8 to uniform C, at least until the nonuniform case is worked out.
[Propositions 4.3 and 4.4] The paper cites Propositions 4.3 and 4.4 from [11] and [10] and then states that 'clearly the proofs given in that paper also work for real matrices.' These propositions are used essentially in Theorem 4.6 and Lemma 4.7 (to establish linear independence of the constructed Gale columns and the existence of a positive vector in the left kernel), and the definitions there are transposed from rows to columns. The cited results may indeed extend to real matrices, but the manuscript should give the precise statements in the cited papers or a short proof sketch, so the reader does not have to reconstruct the transposition.

minor comments (3)

[Proof of Lemma 4.5] In the proof of Lemma 4.5, the sentence 'The matrix D∈ R^{n×k} obtained from \tilde D by adding D0 as a first column vector' should read 'D∈ R^{n×(k+1)}', since \tilde D is n×k and adding one column produces an n×(k+1) matrix.
[Section 6.2, after Theorem 6.10] The sentence 'But note that when rk([A]_2) = d + 1, then 2d−rk([A]_2) = 1/2' should be typeset as an exponent, namely 2^{d−rk([A]_2)} = 1/2, to avoid confusion with the linear expression.
[Example 3.9] In Example 3.9, the phrase 'Then, there exists a vector δ∈ R^2 such that ⟨Pi,δ⟩ > 0' should specify that the vector δ is chosen from the nonempty open cone dual to the P_i, since the existence of such δ is equivalent to the strict convexity of the positive cone generated by the P_i, which follows from 0∈C^∘.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main sufficiency theorems are derived from independent degree-theoretic and Gale-duality ingredients, with self-citations used only as foundational or illustrative tools.

full rationale

The paper's central claim, Theorem 3.8, is not equivalent to its inputs. Existence of a positive root is reduced, via Theorem 2.5 (proved in the paper), to existence of a zero of the Gale map g in the bounded polytope Δ_P; existence is then forced by an inward-vector Brouwer degree criterion (Theorem 3.1, cited from [6,7], not from the authors). Conditions (1) and (2) are explicit sign and non-mixedness constraints on Gale dual matrices B and D; Lemmas 3.4 and Corollary 3.5 are proved directly, and condition (2) makes −g point inward over the boundary. No parameter is fitted to the target quantity n_A(C), and no 'prediction' is a renamed input. The later theorems (4.6, 5.8) use external results of Fischer–Shapiro [11], Fischer–Morris–Shapiro [10], and Peeva–Sturmfels [16]; Theorem 6.10 is proved from the Gale-dual correspondence. Self-citations to the authors' prior work ([2,4,5,13]) are either supplied with proofs in this paper (Theorem 2.5) or used only for illustration and context (Example 3.9 and the introduction); none is the load-bearing argument for the main theorem. One non-circular proof gap should be flagged: in the proof of Theorem 3.8, after defining the reduced map ar g with constants c_j (equation (3.7)), the text asserts without proof that 'It is then not difficult to see that, the conclusions of Lemma 3.4 and Corollary 3.5 hold for the map ar g' and then says 'To simplify the notation, we assume from now on that C is uniform', even though Theorem 3.8 states the nonuniform case. This is an omitted justification about the nonuniform case, not a circular reduction: the constants c_j are positive and do not alter signs, so the assertion is plausibly repairable; nonetheless, as written, the full generality of Theorem 3.8 rests on that unproved extension. This gap does not raise the circularity score because the derivation is not by construction equivalent to the conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard theorems (Brouwer degree, Gale duality, mixed dominating matrices, lattice ideal classifications) imported from the cited literature. No free parameters are fitted to data. The main in-house burden is the unproved extension of the boundary-sign lemmas to the nonuniform Gale map and the asserted real-matrix extension of [11].

assumptions (6)

standard math Gale duality bijection between positive solutions of (1.1) and solutions of the Gale dual system (2.8) in the projectivized dual cone, stated as Theorem 2.5 from [4].
The entire approach transfers the original system to the dual system; the result is cited from Bihan-Sottile rather than proved from first principles.
standard math Inward-pointing vector field criterion of Brouwer degree, stated as Theorem 3.1 from [6,7]: if h is C^1 and h points inward on the boundary of a bounded convex open set, then h has a zero there.
This supplies the topological existence step in Theorem 3.8.
domain assumption Mixed dominating matrix facts: any nonzero linear combination of columns of a mixed dominating matrix is mixed (Proposition 4.3), and its left kernel contains a positive vector (Proposition 4.4).
Theorem 4.6 uses these propositions over the real numbers.
domain assumption Characterization of configurations admitting a mixed dominating Gale dual via the two-part decomposition, stated as Theorem 5.1 from [10].
Used to prove Lemma 5.4, Corollary 5.5 and Theorem 5.8.
domain assumption Uniformity hypotheses on A and C, meaning that all maximal minors are nonzero, in Theorems 4.6, 5.8, 5.10, 5.11 and Proposition 6.4.
Uniformity makes Gale dual matrices uniform and simplifies the face structure; Theorem 3.8 tries to relax it, but its proof skips the nonuniform case.
standard math Peeva-Sturmfels characterization in Proposition 6.3: for k=2, the lattice ideal I_LB is not Cohen-Macaulay if and only if its Gale diagram meets all four open quadrants.
Used to translate the quadrant condition into algebraic language in Proposition 6.4.

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Pith. "Pith review of Sign conditions for the existence of at least one positive solution of a sparse polynomial system." pith.science (2026). https://pith.science/paper/SQDUYTH6

@misc{pith2026190805503,
  author       = {Pith},
  title        = {Pith review of: Sign conditions for the existence of at least one positive solution of a sparse polynomial system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQDUYTH6}},
  note         = {Machine review of arXiv:1908.05503}
}

read the original abstract

We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of integer exponents, we relate our sufficient conditions to algebraic conditions that emerged in the study of toric ideals.

Figures

Figures reproduced from arXiv: 1908.05503 by the authors.

**Figure 1.** The polytope ∆P of Example 3.10, with c > 0. k; k[1]=48c12+1280c11+12288c10+49152c9+65467c8-2560c7-24576c6- 98304c5-131078c4+1280c3+12288c2+49152c+65563 The roots of this last polynomial in c correspond to systems with a degenerate solution, and we can check that the only positive root of f1, f2 and their jacobian is 1. We check, again using Singular [8], with the library “signcond.lib” (implemented by E. Tobis, ba… view at source ↗

**Figure 2.** A and C are I-compatible for I = {1, 2, 3, 4}. We have the following result: Theorem 5.8. Assume that A, C and C¯ are uniform. Suppose 0 ∈ C◦ , and there exists I ⊂ {1, . . . , n} such that A and C are I-compatible. Then, nA(C) > 0. Proof. Let BI be a Gale dual matrix of AI as in Condition (1) of Definition 5.6. As aj ∈ chull(AI ) for each j /∈ I, we can use Lemma 5.4. We construct a dominating matrix B, using the … view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

K. G. Fischer and J. Shapiro. Mixed matrices and binomial ideals. Journal of Pure and Applied Algebra, 113(1):39–54, 1996

work page 1996

[2] [10]

K. G. Fischer, W. Morris, and J. Shapiro. Mixed dominating matrices. Linear algebra and its applications, 270(1-3):191–214, 1998

work page 1998

[3] [1]

S. Basu, R. Pollack, and M.-F. Coste-Roy. Algorithms in Real Algebraic Geometry. Volume 10. Springer Science & Business Media, 2007

work page 2007

[4] [2]

Bihan and A

F. Bihan and A. Dickenstein. Descartes’ rule of signs for polynomial systems supported on circuits. International Mathematics Research Notices, 2017(22):6867–6893, 2016

work page 2017

[5] [3]

Bihan, F

F. Bihan, F. Santos, and P. Spaenlehauer. A polyhedral method for sparse systems with many positive solutions . SIAM Journal on Applied Algebra and Geometry, 2(4):620– 645, 2018

work page 2018

[6] [4]

Bihan and F

F. Bihan and F. Sottile. New fewnomial upper bounds from Gale dual polynomial systems. Moscow Mathematical Journal, 7(3):387–407, 2007

work page 2007

[7] [5]

Bihan and F

F. Bihan and F. Sottile. Gale duality for complete intersections . In Annales de l’institut Fourier, 58(3):877–891, 2008

work page 2008

[8] [6]

Conradi, E

C. Conradi, E. Feliu, M. Mincheva, and C. Wiuf. Identifying parameter regions for multistationarity. PLoS computational biology, 13(10):e1005751, 2017

work page 2017