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New bounds for the support of input-output equations in differential-algebraic systems
Gabriela Jeronimo, Leonardo Lanciano
TL;DR
The paper addresses how to derive minimal input-output differential equations without hidden states, a task for which effective elimination methods are limited by polynomial growth. It develops a Perron-based approach to bound eliminant supports with varying inputs; experiments find sharp support predictions in tested constant-parameter cases but less sharp estimates for differential parameters.
Problem
Effective differential elimination remains limited by rapid growth in polynomial size, while input-output equations seek state-free differential consequences relating measured inputs and outputs.
Source: §1 · p. 2Method
A Perron-theorem analogue provides explicit degree and support bounds for eliminant polynomials with freely varying inputs.
Source: §1 · pp. 2–3Results
In all tested constant-parameter examples, the bounds accurately predict exact eliminant support, while estimates for differential parameters are noticeably less sharp.
Source: §5 · p. 23Takeaways & Limitations
Classical algebraic elimination is severely limited by growing candidate supports and eliminants, while experiments suggest evaluation–interpolation is more suitable for computation.
Source: §5 · p. 23Differential-parameter bounds are less sharp, and increasing candidate supports and eliminant sizes make their asymptotic sharpness computationally challenging to assess.
Source: §4.2 · p. 23, §5 · p. 23
Abstract
from arXiv · showhide
Given a polynomial dynamical system $\mathbf{x}'=\mathbf{f}(\mathbf{x},\mathbf{u})$ together with an observation function $y=g(\mathbf{x},\mathbf{u})$, where $\mathbf{x}=(x_1,\ldots,x_n)$, $\mathbf{u}=(u_1,\ldots,u_m)$ and $y$ are differential variables, and $\mathbf{f}=(f_1,\ldots,f_n)$, $g$ are polynomials with coefficients in a differential field, we study the problem of determining a minimal polynomial differential equation satisfied by the inputs $\mathbf{u}$ and the output $y$ which follows as a differential consequence of the system. We provide a characterization of a finite superset of the set of monomials appearing with non-zero coefficients in this input-output equation. Specifically, we establish an upper bound for the degree of the minimal polynomial and a family of inequalities that define a polytope containing its Newton polytope. These results extend recent work by Mukhina and Pogudin for systems with constant parameters, and enable the use of evaluation-interpolation techniques for the efficient computation of such eliminant polynomials.
1 Introduction
The paper seeks minimal differential equations relating measured inputs and outputs without the hidden state variables. It develops support bounds for these equations when inputs may vary, extending prior work and motivating evaluation–interpolation computation.
1 Introduction
- Input-output equations eliminate unmeasured states, but effective differential-elimination methods remain limited by rapid polynomial growth.Such state-free relations are relevant when inputs and outputs are measurable and arise in applications including parameter identifiability.Source: §1 · pp. 1–2
- The main contribution is explicit support bounds for eliminant polynomials when inputs vary rather than remain constant.The theorem is formulated using degrees of the observation function and system polynomials.Source: §1 · p. 2
- The Perron-theorem analogue provides a unified approach that recovers most existing bounds and yields new ones without constant-parameter counterparts.The result generalizes [22, Theorem 1]; its type-4 inequalities are new, and its constant-parameter bounds are slightly sharper than prior bounds.Source: §1 · p. 3
- The candidate support extends Mukhina and Pogudin’s evaluation–interpolation procedure to differential parameters.The paper also reports numerical experiments assessing the bounds.Source: §1 · pp. 2–3
2 Preliminaries
The preliminaries establish notation for differential polynomials and introduce geometric degree and height tools. A generalized Perron-type result uses these tools to bound supports when a polynomial map has hypersurface image.
2.1 Basic definitions and notation
- Differential polynomials in x and u form a polynomial ring generated by their successive derivatives.The differential ideal [H1, …, Hβ] is the smallest ideal containing its generators and closed under derivation.Source: §2.1 · pp. 3–4
- The total derivative combines coefficient derivation with partial derivatives in the differential variables.Higher total derivatives are denoted F^(q).Source: §2.1 · p. 4
- The support records a polynomial’s monomials, and its Newton polytope is their convex hull.The polytope lies in the real exponent space for the variables under consideration.Source: §2.1 · p. 4
2.2 Some auxiliary facts from algebraic geometry
- Geometric degree measures complexity in x, while height measures complexity in parameter variables.For varieties over a function field, height can be defined through the parameter-degree of the Chow form.Source: §2.2 · p. 5
- The generalized Perron result bounds supports of a primitive irreducible polynomial when the map’s image is a hypersurface.The bounds involve the variety’s height and degree and upper bounds on the coordinate polynomials’ degrees.Source: §2.2 · pp. 5–6
- The proof uses a generically finite map to a hypersurface and identifies its defining equation with the polynomial to which the support bounds apply.The argument invokes a result allowing degree upper bounds in place of exact degrees.Source: §2.2 · p. 6
3 Bounds for the Support of the Eliminant Polynomial
This section develops bounds that constrain the support of the eliminant polynomial, including degree and exponent bounds and constraints on fixed-order parameter derivatives. It also presents refined estimates that use output structure through partial evaluation.
3.1 Eliminant polynomials
- The eliminant is the unique irreducible equation after eliminating x; its order ν is the first algebraic dependence among y, …, y(ν), and the two evaluation-map images define the same hypersurface.The primitive defining equations coincide up to a scalar factor.Source: §3.1 · pp. 8–9
3.2 Degree bounds and proof of the main result
- Each differentiation-evaluation step raises x-degree by at most Dx − 1 and u(≤ν)-degree by at most Du; the total-degree bound is deg(E(g(i))) ≤ d + i(D − 1).The corresponding x-degree bound is degx(E(g(i))) ≤ dx + i(Dx − 1).Source: §3.2 · pp. 10–12
- Theorem 3.7 bounds the eliminant’s degree and gives inequalities constraining the exponents of monomials in its support.For the example, the inequalities include ℓ0 + ℓ1 + ℓ2 + 2e0 + 2e1 + 2e2 ≤ 8 and ℓ0 + ℓ1 + ℓ2 + 2e0 + 3e1 + 4e2 ≤ 9.Source: §3.2 · pp. 11, 13, 14
- Unlike the separate x- and u-degree bounds in, the total-degree approach makes the analogous inequality nonredundant and strictly stronger in general.The cited bounds concern constant parameters, where one of their inequalities is redundant.Source: §3.2 · p. 13
- 140 lattice points satisfy the example’s five nonredundant inequalities, while its actual eliminant has 10 terms, showing that the polytope can substantially overestimate the support.The example also shows that each inequality is nonredundant.Source: §3.2 · p. 14
- Theorem 3.7 adds bounds on fixed-order parameter derivatives, with at least one sharper than earlier inequalities when r = ν and either dx > 1 or Dx > 1.For fixed 1 ≤ r ≤ ν, the inequalities constrain the degree in u^(r).Source: §3.2 · p. 14
- Theorem 3.7 also applies to non-autonomous polynomial systems by adjoining time as a state variable with derivative 1.This allows the bounds to account for polynomial degrees in t.Source: §3.2 · p. 14
3.3 Exploiting the structure of the output: partial evaluation
- Full evaluation can overestimate degrees because it treats derivatives uniformly and misses cancellations, sparsity, and other structure after substitution.For outputs y = h(x1), the derivatives have a specific structure that enables sharper weighted bounds.Source: §3.3 · p. 15
- For outputs y = h(x1), Corollary 3.11 derives refined bounds directly from output structure, while partial evaluation yields further inequalities by retaining selected derivatives as independent variables.The corollary assumes d > 0 and δx > 0; the partial-evaluation method sharpens estimates when δx exceeds the state-degree contributions of the other equations.Source: §3.3 · pp. 15–16
- No inequality in Proposition 3.15 can be removed: Examples A and B show that every stated bound is non-redundant.The examples use ν = n = 2 and test the effect of removing each inequality in turn.Source: §3.3 · p. 19
- For n = ν = 2, the nonparametric bounds (3) and (4) are valid when δx > Dx ≥ 0; they recover the generic Newton polytope when Dx = d = 1 and sharply approximate it when d = 1.The generic-system result is identified as [21, Theorem 2].Source: §3.3 · p. 19
4 Experimental results
The experiments assess the support bounds and reconstruction procedure, finding exact support predictions in tested constant-parameter examples but less sharp bounds with differential parameters.
4.1 Algorithm and strategy
- The evaluation–interpolation implementation constructs candidate support from the degree bounds and reconstructs the eliminant by interpolation over a finite field.It extends Mukhina and Pogudin’s procedure to differential parameters and was implemented in Julia.Source: §4.1 · p. 20
4.2 Computations
- 1292 monomials match the exact eliminant support versus 12007 in DiffMinPoly for one constant parameter; with two, 8189 match exact support versus 437228.The reported gap between candidate supports increases further as the number of parameters grows.Source: §4.2 · p. 21
- In sufficiently dense constant-parameter systems with dx = du = d and Dx = Du = D, the Theorem 3.7 inequality appears to describe the eliminant’s Newton polytope exactly.The candidate-support size directly determines the square interpolation matrix size, so smaller supports yield smaller interpolation problems.Source: §4.2 · p. 22
- With differential parameters, input derivatives add variables to the eliminant, accelerating candidate-support growth and making reconstruction more challenging.The experiments assess both the effectiveness and limitations of the bounds in this setting.Source: §4.2 · p. 22
- The differential-parameter bounds are less sharp than the constant-parameter bounds, and growing supports and eliminants make their asymptotic sharpness difficult to assess.Experiments construct systems with n = 2 and report support sizes stable across generic system choices and increasing primes.Source: §4.2 · pp. 21, 23
5 Conclusions and future work
The paper develops support bounds for input-output equations with differential inputs and evaluates their computational use. The bounds are accurate in tested constant-parameter cases, while differential parameters and omitted sparsity patterns leave important scope for improvement.
5 Conclusions and future work
- The paper gives the first support descriptions for input-output equations of differential-algebraic systems with differential inputs, using a Perron-theorem analogue based on variety height.The approach recovers prior Mukhina–Pogudin results and extends them to differential parameters.Source: §5 · p. 23
- Computational experiments indicate that evaluation–interpolation is more suitable than classical algebraic elimination here, whose candidate supports and eliminants grow rapidly.Source: §5 · p. 23
- In all tested constant-parameter examples, the bounds accurately predict exact support; differential-parameter estimates are noticeably less sharp, with derivative structure offered as an explanation.The authors expect more symmetric treatment of parameters and their derivatives might yield behavior closer to the constant-parameter case.Source: §5 · p. 23
- More accurate support descriptions for differential-parameter eliminants remain an open question.Source: §5 · p. 24
- The bounds use degrees but not sparsity patterns; sharper general results require analyzing sparse systems and the structure of successive derivatives.Source: §5 · p. 24
- With differential inputs, rapidly growing candidate supports enlarge interpolation matrices until their size becomes the main computational bottleneck.Further algorithmic progress is described as requiring both sharper bounds and more refined evaluation–interpolation strategies.Source: §5 · p. 24