Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations

Boris Kramer; Gleb Pogudin

REVIEW 1 major objections 6 minor 69 references

Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations

T0 review · 1 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Most nonlinear ODE models can be recast as quadratic systems by adding new variables, this chapter argues.

desk verdict Solid survey, not a research paper: the central existence theorems hold up, but Example 4 has a real chain-rule error that needs fixing before this is citable as a reference. read the letter →

arxiv 2502.10005 v1 pith:HJDTHKJ2 submitted 2025-02-14 cs.SC cs.NAmath.DSmath.NAq-bio.MN

Boris Kramer , Gleb Pogudin This is my paper · ORCID

classification cs.SCcs.NAmath.DSmath.NAq-bio.MN MSC 34-0234C2068W30

keywords polynomializationquadratizationliftingtransformationsnonlinearODEsdifferentially-algebraicfunctionsmonomialsymboliccomputationchemicalreactionnetworks

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 chapter argues that polynomialization and quadratization—adding auxiliary variables so that an ODE system's right-hand side becomes polynomial, then quadratic—are broadly available and practically useful transformations for nonlinear models. The paper's central existence results state that any system whose right-hand side is made of differentially-algebraic functions (functions that satisfy an algebraic differential equation, such as exponentials, sines, and logarithms) can be polynomialized, and that every polynomial system admits a monomial quadratization whose order is bounded by a product of one plus the per-variable degrees. On the algorithmic side, the paper reviews two software tools that can discover these representations automatically, with the caveat that current polynomialization tools offer no optimality guarantee. If these claims hold, many difficult nonlinear models become accessible to the standard quadratic-system toolbox used in simulation, model reduction, reachability analysis, and chemical-reaction-network computing.

What carries the argument

The central objects are the lifting transformation and the differential-algebraicity condition. A polynomialization is a map $x \mapsto [x, w(x)]$ with $w$ a vector of new functions chosen so that the augmented system has polynomial right-hand sides; a quadratization is the same idea with degree at most two, and in the monomial case the new entries are monomials. The machine that makes existence work is the class of differentially-algebraic functions, which is closed under arithmetic and composition; Theorem 1's proof is constructive, and for quadratization the construction is explicit: take the finite set $M$ of all monomials $x_1^{e_1}\cdots x_n^{e_n}$ with $0 \le e_i \le d_i$, and the derivative of any $m \in M$ is quadratic in $M$. Algorithmically, the search is carried out by either SAT-solving over this large monomial set or tree search over candidate variable sets.

What would settle it

Find or construct a polynomial ODE system in $n$ variables whose optimal monomial quadratization requires a number of new variables that grows faster than any polynomial in $n$; the paper notes that Conjecture 1 says such systems exist, so exhibiting one with a certified exponential lower bound would show that low-order quadratization cannot be expected for all polynomial systems.

Watch

Extended reading notes

Core claim

On the paper's own terms, the core discovery is a pair of existence theorems with practical reach. Theorem 1 says that if the functions $f(x)$ are analytic and differentially-algebraic with respect to $x$ on a domain $D$, then system $\dot{x}=f(x)$ has a polynomialization: there are new variables $w(x)$ and polynomials $q_1,q_2$ such that trajectories of the original system are mapped to trajectories of the polynomial system $\dot{x}=q_1(x,w)$, $\dot{w}=q_2(x,w)$. Theorem 2 says that every polynomial ODE system $\dot{x}=p(x)$ has a monomial quadratization, with an optimal monomial quadratization of order at most $\prod_{i=1}^n(d_i+1)$, where $d_i$ is the degree of $p$ with respect to $x_i$; the proof takes all monomials with exponents bounded by $d_i$ as new variables. Together these theorems turn 'look for hidden polynomial or quadratic structure' from an art into a search problem with existence guarantees, and the chapter documents that these guarantees are already implemented in practice for a useful class of elementary functions.

Load-bearing premise

The load-bearing premise is that the system's right-hand side is smooth and differentially-algebraic on the domain of interest; if a model contains a non-differentially-algebraic or non-smooth term, such as a piecewise or data-defined function, the existence of a polynomialization and the software guarantees do not apply.

Editorial extensions

If this is right

Every polynomial ODE has a monomial quadratization of finite order, so the question of whether a degree reduction is possible is settled; the remaining question is how small the order can be.
Polynomialization turns models built from exponentials, sines, logarithms, and compositions of them into polynomial systems, opening them to standard quadratic-system analysis.
Quadratized systems can be interpreted as chemical reaction networks with bimolecular reactions, which is a step toward analog and Turing-complete chemical computing.
A non-autonomous polynomial ODE with differentiable inputs admits a monomial quadratization; if the input is not differentiable, an input-free quadratization may fail, so differentiability is a real boundary.
For applications like a sigmoid perceptron and the MAPK cell-signaling model, automatic quadratization produces compact systems and even reveals closed subsystems that are not obvious from the original equations.

Reading between the lines

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

A straightforward extension would be to let users of polynomialization software supply a special function by its defining differential equation, such as the Bessel function $J_\alpha$; the paper points to this idea but does not implement it.
If optimal quadratization is NP-hard in general, then in practice one should expect a trade-off between expressive new-variable languages (Laurent monomials, arbitrary polynomials) and tractability; the paper notes these can reduce dimension but no general algorithm exploits them.
For data-driven model reduction, the existence of low-order quadratizations suggests that learned latent coordinates might be constrained to monomial or polynomial functions rather than free-form neural-network coordinates; that connection is implicit but not developed.
The same lifting idea could be applied to partial differential equations, but the chapter only cites preliminary ODE-based results; a testable next step is to quadratize spatially discretized PDEs and compare the resulting reduced-order models with direct discretization.

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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

The one thing to know: this is a review chapter, not a new-results paper. It collects the known existence theory for polynomialization and quadratization of ODEs, describes the two main software tools (BioCham and QBee), and works through a few instructive examples. The advertised value is real: if you work in model reduction, chemical reaction networks, or control, this is a usable entry point into that literature, and Section 5's open problems are honestly framed.

What's genuinely good: the proof of Theorem 2 (monomial quadratization always exists) is clear and self-contained, and the examples that I checked—Exponent, the sigmoid perceptron, the suboptimality example—are algebraically correct. The authors are also honest about what is and isn't algorithmic, and they flag the manual step in the perceptron reduction rather than hiding it. That's decent scholarship.

Now the soft spots, in proportion. First, Example 4, the Bessel function polynomialization, has a genuine mathematical error. The new variables are written as functions of t, and their derivatives omit the chain-rule factor from dx/dt. It's a localized defect in an illustrative example, and the intended point—that Bessel functions can be handled if you know their differential equation—survives if the variables are defined as functions of x and the chain rule is applied. But as printed, the ODE system for w is wrong, and a reader who uses it as a template will get confused. The stress-test note agrees this is an editing defect, not a load-bearing flaw, and I concur.

Second, the MAPK quadratization (15 new variables) is reported as a QBee computation, but the quadratic system itself is omitted and no reproducible artifacts are provided. It's not central to the paper, but for a survey that wants to showcase software, this is a missed opportunity—the reader has to take the claim on faith. Third, the paper leans heavily on the authors' own prior work (QBee, the non-autonomous quadratization results). That's not a flaw by itself, but it means the survey's selection of algorithms is to some extent a map of their own research program.

Net verdict: the central existence claims are sound, the writing is clear, and the errors are localized. This deserves a serious referee, but only after the Bessel example is corrected and ideally with the MAPK computation made reproducible. I'd send it to review, but with a clear request for revision, not as a straight accept.

For your reading group: maybe. It's a useful survey to skim, but there's no new theorem or algorithm to sink your teeth into.

Referee Report

1 major / 6 minor

Summary. This chapter surveys the theory and algorithms for converting finite-dimensional ODE systems into polynomial or quadratic form by adjoining new variables. It defines polynomialization and quadratization, states existence results for differentially-algebraic right-hand sides and for polynomial systems, proves a bound for monomial quadratization (Theorem 2), describes the BioCham and QBee software, and illustrates the methods on a sigmoid perceptron model and a MAPK signaling model. The chapter closes with a list of open problems.

Significance. The central existence guarantees are useful and, as far as the surveyed derivations go, sound. The self-contained proof of Theorem 2 is elementary and correct, and most examples verify their constructions explicitly. The chapter is valuable in bringing together results from symbolic computation, control, and systems biology and in pointing to concrete software implementations. The reduction of a sigmoid perceptron model to a two-dimensional quadratic system is a particularly effective demonstration. However, the Bessel-function example contains a mathematical error that must be corrected before the chapter can serve as a reliable reference, and the MAPK claim would benefit from a reproducible output.

major comments (1)

[Section 2.3, Example 4] The displayed polynomial system in Example 4 is not a polynomialization of dot x = J_alpha(x) in the sense of Definition 1. If the new variables are w1 = 1/x, w2 = J_alpha(x), w3 = J'_alpha(x), then along solutions the correct derivatives are dot w1 = -w1^2 w2, dot w2 = w2 w3, and dot w3 = (-w1 w3 + (alpha w1^2 - 1)w2) w2, because each derivative must carry the chain-rule factor dot x = w2. The system printed in the manuscript (with dot w1 = -w1^2, dot w2 = w3, dot w3 = -w1 w3 + (alpha w1^2 - 1)w2) is instead valid for an auxiliary variable t satisfying dot t = 1, after conflating the state x with the independent variable of Bessel's equation. Please correct the example; the intended point survives, since the corrected system is still polynomial.

minor comments (6)

[Section 1.4] The symbol for the nonnegative integers is printed as "Z/greaterorequalslant0"; this is a corrupted encoding of \mathbb{Z}_{\ge 0} and should be fixed.
[Section 2.2, Definition 2] The sentence "there exists and integer h" should read "there exists an integer h", and it may be worth specifying that h is nonnegative.
[Section 3.2, Theorem 4] The bound written as "\Pi^{n+r}_{i=1}(d_i+1)" should be typeset as a product over i from 1 to n+r, and the definition of d_{n+i} should clarify that these are the degrees in the input variables u_i.
[Section 4.1] The reconstruction formula x = -1/a ln(w4/w1^2) - b/a assumes a is nonzero and that the logarithm argument is positive; these assumptions should be stated explicitly.
[Section 4.2] The 15-variable quadratization of the MAPK model is reported without the resulting quadratic system, so the computation cannot be checked from the chapter; please include the full system in an appendix or provide the QBee output and version in the text.
[Section 5] The phrase "These/emdash.cyrpossibly incomplete/emdash.cyrresearch questions" appears to be a corrupted encoding of "These—possibly incomplete—research questions" and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity; core theorems are external or self-contained, and the only defect is a localized chain-rule slip in Example 4.

full rationale

The paper is a survey, not a derivation that consumes its own conclusion. Theorem 2 is proved in full in Section 3.2: the monomial set M is constructed from the stated degrees d_i, every monomial of p(x) lies in M, and the derivative of each m in M is a quadratic polynomial in elements of M, so the quadratization is genuinely built from the input system. Theorem 1 is attributed to Hernández-Bermejo and Fairén [33], an external source, and it is not a restatement of Definition 2 by construction: the definition only asserts differential-algebraicity of f, while the theorem supplies a finite set of new variables whose time derivatives are polynomial, which is a substantive classical result. The other cited existence results, including Theorem 4 from the authors' own prior work, are presented as literature results with attribution, not derived circularly from the present chapter, and they do not fit any parameter to data or rename an input as a prediction. The only concrete defect found is in Example 4, where the derivatives of w2 and w3 omit the chain-rule factor x-dot; this is a localized editing slip that does not affect the main existence theorems or the proof of Theorem 2. Overall, no step in the claimed derivation chain reduces to its own inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The survey's central claims rest on standard results about differentially-algebraic functions and on polynomial and quadratization existence theorems, on domain assumptions about differentiable inputs and analytic right-hand sides, and on trust in the publicly cited software implementations. No new physical or mathematical entities are introduced beyond the lifting variables w, which are standard auxiliary functions. The chapter introduces no fitted free parameters; all constants in examples are symbolic model parameters.

assumptions (4)

domain assumption The right-hand sides f considered for polynomialization are analytic and differentially-algebraic with respect to each state variable on a domain D.
Invoked in Definition 2 and Theorem 1 (Section 2.2). The survey's claim that 'the majority' of nonpolynomial systems can be polynomialized relies on the identification of real-world models with the class of differentially-algebraic functions. This excludes, for example, non-analytic or piecewise-defined systems that lack polynomial differential equations.
domain assumption Input signals u(t) in non-autonomous quadratization are differentiable (or satisfy the conditions of [13, Proposition 3.7] for input-free quadratization).
Theorem 4 (Section 3.2) guarantees a monomial quadratization only under differentiability of inputs; Example 6 shows an input-free quadratization fails without this condition.
standard math Ostrowski's Proposition 1: sums, products, quotients, and compositions of differentially-algebraic functions are differentially-algebraic.
Used in Section 2.2 to demonstrate that expressions built from exp, log, trig, etc. are differentially-algebraic, which underpins Theorem 1. The result is cited from [59] and accepted without proof in the chapter.
domain assumption The software tools QBee and BioCham correctly implement the surveyed polynomialization and quadratization algorithms and produce the outputs claimed in Sections 4.1 and 4.2.
The chapter's application examples rely on 'computation with QBee shows...' and 'Both BioCham and QBee find...'. No code or output files are provided, so the reader must trust the referenced implementations, with the caveat that Example 3 demonstrates they are not optimal.

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Cite this review

Pith. "Pith review of Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations." pith.science (2026). https://pith.science/paper/HJDTHKJ2

@misc{pith2026250210005,
  author       = {Pith},
  title        = {Pith review of: Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJDTHKJ2}},
  note         = {Machine review of arXiv:2502.10005}
}

read the original abstract

Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that the majority of nonpolynomial nonlinear systems can be recast in polynomial form, and their degree can be reduced further to quadratic. This process of polynomialization/quadratization reveals new variables (in most cases, additional variables have to be added to achieve this) in which the system dynamics adhere to that specific form, which leads us to discover new structures of a model. This chapter summarizes the state of the art for the discovery of polynomial and quadratic representations of finite-dimensional dynamical systems. We review known existence results, discuss the two prevalent algorithms for automating the discovery process, and give examples in form of a single-layer neural network and a phenomenological model of cell signaling.

Discussion (0). Continue with ORCID to comment.

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