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Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry

Stefan Müller, Elisenda Feliu, Georg Regensburger, Carsten Conradi, Anne Shiu, Alicia Dickenstein

arXiv:1311.5493v2math.AGmath.DS

TL;DR

Deciding whether positive solutions are unique is difficult in general. The paper characterizes injectivity using sign vectors and proves a partial multivariate generalization of Descartes’ rule.

  • Problem

    Deciding uniqueness of positive solutions is difficult in general, although it is desirable in many settings.

    Source: §1 · p. 2

  • Method

    The paper characterizes injectivity of polynomial-map families with arbitrary real exponents using sign vectors, extending conditions based on Jacobians and determinants.

    Source: §1 · p. 2

  • Results

    The framework proves a partial multivariate generalization of Descartes’ rule, including a bound on positive solutions and an existence statement.

    Source: §1.3 · p. 6, §3.3 · p. 19

  • Takeaways & Limitations

    Injectivity precludes multiple solutions, making it useful when uniqueness of positive solutions is desired.

    Source: §1 · p. 2

    Without further restrictions on the exponent matrix, a power-law system may behave in physically or chemically meaningless ways, including reaching the positive-orthant boundary in finite time with nonzero velocity.

    Source: §3.1 · p. 14

Abstract

from arXiv · show

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry,our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes' rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

1 Introduction

Injectivity can rule out multiple solutions, but deciding uniqueness is difficult in general. The paper develops sign-vector criteria for families of generalized polynomial maps and applies them to chemical reaction networks and real algebraic geometry.

1 Introduction

  • The paper relates and extends existing injectivity conditions expressed through Jacobian matrices and determinants.Its motivation is that injective maps preclude multiple solutions.Source: §1 · p. 2

1.1 Statement of the main theorem

  • The main theorem characterizes family-wide injectivity with respect to a set through sign vectors associated with ker(A) and im(B).For fixed B, its condition depends only on the sign vectors of ker(A) and the set; injectivity also holds with respect to the largest set having the same sign vectors.Source: §1.1 · p. 3

1.2 Motivation from chemical reaction networks

  • For chemical reaction networks, injectivity with respect to the stoichiometric subspace rules out multistationarity, and the paper extends criteria for injectivity and steady-state multiplicity.The paper also gives conditions for precluding multistationarity and for guaranteeing multiple special steady states.Source: §1.2 · p. 4, §1.3 · p. 7

1.3 Application to real algebraic geometry

  • Theorem 1.5 gives an at-most-one-positive-solution condition and an existence-and-uniqueness condition for generalized polynomial systems.The univariate case recovers the corresponding one-positive-root conclusion of Descartes’ rule, including for real exponents; the paper presents the theorem as a partial multivariate generalization depending on coefficient and monomial signs.Source: §1.3 · pp. 5–6

2 Sign conditions for injectivity

This section develops sign-vector criteria for injectivity of generalized polynomial maps with positive parameters, relating them to Jacobian matrices and determinant-based conditions, including equivalent determinant and minor-sign criteria specialized to constraint subspaces of the form im(A).

2.2 Families of linear maps

  • For a vector subspace S, ker(AκBλ)∩S* = ∅ reduces to ker(AκBλ)∩S = {0}; equivalently, AκBλ is injective on S.The statements in Lemma 2.1 are necessary conditions for the statements in Proposition 2.2.Source: §2.2 · p. 8

2.3 Families of generalized monomial/polynomial maps

  • For generalized monomial and polynomial maps, injectivity with respect to a subset is characterized by sign-vector conditions, extending the linear-family criteria to maps of the form fκ(x) = Aκ xB.For a vector subspace, monomial-map injectivity on each positive-orthant coset is equivalent to injectivity of Bλ on that subspace for every λ.Source: §2.3 · p. 9
  • The Jacobian matrices of fκ(x) = Aκ xB coincide with the matrices AκBλ, linking the sign and injectivity conditions to Jacobian-based criteria.The proof of Theorem 1.4 establishes equivalence among its linear, sign, injectivity, and Jacobian statements.Source: §2.3 · p. 10
  • Injectivity on the positive orthant also constrains boundary extensions: under the stated coordinate condition, distinct points in the same coset have distinct images for every parameter choice.The condition requires that simultaneous vanishing of corresponding monomials entails zeros in every coordinate with a nonzero exponent; in particular, an interior and a boundary point in one coset cannot share an image.Source: §2.3 · p. 11

2.4 Determinantal conditions

  • When the subspace dimension equals rank(A), the section gives injectivity conditions in terms of determinants and signs of maximal minors.This subsection begins by setting up relations between maximal minors of matrices representing a subspace and its orthogonal complement.Source: §2.4 · p. 11
  • For the constrained map, injectivity is equivalent to det(Γκ,λ) being nonzero with same-sign nonzero coefficients, or all s-minor products being zero or sharing a sign, with at least one nonzero.Source: §2.4 · pp. 12–13
  • The test can use either the nonzero s × s minor products or symbolic det(Γκ,λ), which is the sum of size-s principal minors of eAκBλ.When det(Γκ,λ) is nonzero, it equals the product of the nonzero eigenvalues of eAκBλ.Source: §2.4 · p. 13
  • When S = im(A), injectivity is equivalent to the products det(A_I,J) det(B_J,I) being zero or sharing a sign, with at least one nonzero.Source: §2.4 · p. 13
  • For unrestricted injectivity with full-rank A and B, the ker(A)-im(B) sign-vector intersection criterion applies; m < n precludes injectivity, m = n has the stated n-minor sign test, and m > n is NP-complete.Source: §2.4 · p. 13

3 Applications

The applications use sign-based injectivity results to analyze multiple steady states in power-law systems and existence and uniqueness of positive solutions of generalized polynomial equations. They also express an injectivity condition in oriented-matroid language.

3.1 Power-law systems

  • For power-law systems, the sign condition (sig) is equivalent to Jacobian injectivity on the stoichiometric subspace and sign agreement of nonzero minor products, precluding multiple positive steady states per compatibility class for any rate constants.The criteria restate the general injectivity result in the power-law setting.Source: §3.1 · pp. 14–16

3.2 Precluding/guaranteeing special steady states

  • Injectivity of an associated polynomial map is equivalent to uniqueness of special steady states per compatibility class, giving sign conditions to preclude multiplicity.The cases include complex balancing equilibria and toric steady states, whose varieties admit generalized binomial and monomial presentations.Source: §3.2 · pp. 16–18
  • When the sign condition fails and every point is a special steady state for some rate constants, multiple special steady states occur in a compatibility class.For complex balancing, weak reversibility supplies the relevant assumption; for toric steady states, it follows from existence of a positive toric steady state, equivalently a positive vector in ker(N).Source: §3.2 · p. 18

3.3 Solving systems of generalized polynomial equations

  • The sign-based injectivity framework gives a bound on positive solutions that forms a partial multivariate generalization of Descartes’ rule, within an oriented-matroid framework.The bound follows from the injectivity criteria; the existence result relies on a surjectivity result.Source: §3.3 · p. 19
  • The injectivity sign condition holds exactly when no nonzero vector of A is orthogonal to every vector of B^T, equivalently no such covector of B^T is orthogonal to all covectors of A.Source: §3.3 · p. 20
  • Under (ex), injectivity combined with prior surjectivity guarantees a unique positive solution, restating a generalization of Birch’s theorem [53, Proposition 3.9].The surjectivity result used is [53, Theorem 3.8].Source: §3.3 · p. 20
  • The result can fix a full-rank exponent matrix B with rows in an open half-space and use any coefficient matrix A that defines the same oriented matroid as B^T.Source: §3.3 · p. 20
  • When existence is guaranteed independently, Proposition 3.12’s sign condition ensures a unique positive solution; homotopy continuation can find it, but the followed solution may cease to be positive.A starting system can be chosen with x = (1, . . . , 1) as its unique solution.Source: §3.3 · pp. 20–21

4 Algorithmic verification of sign conditions

The sign condition can be checked by testing whether specified linear systems are feasible, with exact linear programming and oriented-matroid methods offering verification routes. Special cases connect the test to maximal minors, while unrestricted cases can be computationally hard.

4 Algorithmic verification of sign conditions

  • Checking injectivity reduces to testing whether any sign-vector pair in the supplied set makes the associated linear system feasible.The matrices A and B are assumed rational, and the method handles a given set of sign vectors T for a union of orthants.Source: §4 · p. 21
  • Homogeneity lets strict sign inequalities be replaced by bounds using any positive ε, enabling exact linear programming and rational infeasibility certificates.The alternative approach may require testing 3r|T| choices of sign-vector pairs.Source: §4 · p. 21
  • For subspaces, sign vectors are oriented-matroid covectors; cocircuits generate them, but covector counts can be exponential, such as 3^n versus n for R^n.An output-sensitive polynomial-time covector-generation algorithm is discussed in.Source: §4 · pp. 21–22
  • In unrestricted injectivity, the test asks whether ker(A) and im(B) share a nonzero covector; when m = n it has a maximal-minor characterization.For integer matrices with m > n, the problem is strongly NP-complete [16, Theorem 5.5]; a determinant-and-sign characterization also applies when dim(S) = rank(A).Source: §4 · pp. 21–22
  • Available tools include mixed-integer branch-and-bound for sign-vector search, Topcom for chirotopes and cocircuits, and RAGLib for real polynomial feasibility.The sign-vector search software has been applied to establish multistationarity in Systems Biology models.Source: §4 · p. 22
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