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arxiv: 2501.13680 · v4 · submitted 2025-01-23 · 💻 cs.SC · math.AG· math.CA

Projecting dynamical systems via a support bound

Yulia Mukhina , Gleb Pogudin This is my paper

Pith reviewed 2026-05-23 05:24 UTC · model grok-4.3

classification 💻 cs.SC math.AGmath.CA
keywords Newton polytopedifferential eliminationpolynomial dynamical systemsminimal differential equationevaluation-interpolationsymbolic computationprojection
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The pith

A bound on the Newton polytope of the minimal differential equation for a coordinate in a polynomial dynamical system depends only on dimension and the input degrees d and D.

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

The paper addresses the problem of projecting a polynomial dynamical system onto one chosen coordinate by computing the minimal differential equation it satisfies. It establishes an explicit bound on the Newton polytope of this minimal equation, expressed in terms of the system dimension together with the degrees d and D of the polynomials governing the distinguished coordinate and the remaining variables. The bound is shown to be tight whenever d is at most D or the system is two-dimensional. The authors then convert the bound into an evaluation-interpolation algorithm that computes the minimal equation and solves instances beyond the reach of existing differential-elimination software.

Core claim

For a polynomial dynamical system, a bound is given for the Newton polytope of the minimal differential equation satisfied by a chosen coordinate. The bound depends on the dimension of the model and the degrees d and D of the polynomials defining the dynamics of the chosen coordinate and the remaining coordinates. The bound is sharp if d ≤ D or the model is planar. This bound is used to design an algorithm for computing the minimal equation via the evaluation-interpolation paradigm.

What carries the argument

The explicit bound on the Newton polytope of the minimal equation, which directly determines the monomials needed in the evaluation-interpolation procedure.

If this is right

  • The minimal equation can be recovered by evaluating the system at sufficiently many points and solving a linear algebra problem whose size is controlled by the bound.
  • Instances that exceed the capacity of current differential-elimination tools become tractable.
  • When d ≤ D or the system is planar the computed equation is guaranteed to have the smallest possible support.
  • The same bound supplies an a-priori estimate of the computational cost of the projection step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The support bound may be reusable as a complexity measure for other coordinate-projection tasks in algebraic differential systems.
  • In modeling applications the bound offers a way to decide in advance whether projecting a given coordinate is feasible before running the algorithm.
  • The evaluation-interpolation strategy could be combined with sparse interpolation techniques to further reduce the number of required sample points.

Load-bearing premise

The input consists of polynomial vector fields whose degrees are exactly d for the distinguished coordinate and D for the others.

What would settle it

A concrete polynomial dynamical system with d > D in dimension greater than two whose minimal equation has a monomial outside the predicted Newton polytope.

read the original abstract

For a polynomial dynamical system, we study the problem of computing the minimal differential equation satisfied by a chosen coordinate (in other words, projecting the system on the coordinate). This problem can be viewed as a special case of the general elimination problem for systems of differential equations and appears in applications to modeling and control. We give a bound for the Newton polytope of such minimal equation. Our bound depends on the dimension of the model and the degrees $d$ and $D$ of the polynomials defining the dynamics of the chosen coordinate and the remaining coordinates, respectively. We show that our bound is sharp if $d \leqslant D$ or the model is planar. We further use this bound to design an algorithm for computing the minimal equation following the evaluation-interpolation paradigm. We demonstrate that our implementation of the algorithm can tackle problems which are out of reach for the state-of-the-art software for differential elimination.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the projection of a polynomial dynamical system onto one coordinate by computing the minimal annihilating differential equation satisfied by that coordinate. It derives an explicit upper bound on the Newton polytope of this minimal equation that depends only on the ambient dimension n and the input degrees d (for the distinguished coordinate) and D (for the remaining coordinates). The bound is shown to be sharp when d ≤ D or when n=2, and is then used to certify an evaluation-interpolation algorithm whose implementation is reported to solve instances beyond the reach of existing differential-elimination software.

Significance. If the stated bound and its sharpness hold, the work supplies a parameter-free support bound that directly enables a certified, practical algorithm in the evaluation-interpolation paradigm for differential elimination. This is a concrete advance for symbolic computation, with clear relevance to modeling and control applications. The explicit dependence solely on n, d, and D, together with the sharpness results and the reported computational gains, constitute the main strengths.

minor comments (2)
  1. [Abstract] The explicit form of the bound (presumably stated in §3 or §4) is described rather than displayed in the abstract; adding the formula would make the central contribution immediately visible to readers.
  2. [§6] Section 6 benchmark tables would benefit from explicit reporting of hardware specifications, timeout values, and memory limits to allow direct reproduction of the claimed performance advantage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; bound derived from input parameters

full rationale

The central result is an explicit upper bound on the Newton polytope of the minimal annihilating equation, stated to depend only on ambient dimension n and the exact input degrees d (distinguished coordinate) and D (remaining coordinates). The paper proves containment, establishes sharpness when d ≤ D or n=2, and applies the bound to an evaluation-interpolation algorithm whose correctness follows directly from the support containment. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the derivation is self-contained against the stated polynomial vector-field inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced or required by the abstract; the work operates within standard algebraic-geometry notions of Newton polytopes and differential elimination.

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discussion (0)

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