Iterative Polynomial Regularisation

Updated 17 October 2025

Iterative polynomial regularisation is a method that systematically transforms polynomial systems to remove singularities and enforce stability through repeated blow-ups and coordinate changes.
The process leverages algebraic and analytical tools such as EM-like descent algorithms and Newton polygon minimality to achieve regular and minimal representations.
This framework unifies approaches in numerical approximation and algebraic geometry, enhancing analyses of nonlinear, imprecise, or ill-posed polynomial models.

The iterative process of polynomial regularisation is a family of algorithmic schemes in which polynomial equations, polynomial systems, or polynomial models are regularised—typically to remove singularities, enforce stability, or construct minimal representatives—by systematically applying repeated transformations, updates, or coordinate changes. This process is especially prominent in settings involving nonlinear or ill-posed problems, high-order Hamiltonian systems, sums of squares decompositions, and numerical approximation with polynomials, and it leverages both analytical and geometrical tools such as EM-like algorithms, optimization-based descent, blow-ups, and transformations tied to Newton polygons and birational geometry.

1. Algebraic and Analytical Foundations

At the heart of iterative polynomial regularisation are polynomial systems or models that exhibit singularities, indeterminacies, or instability under direct analysis or computation. Core examples include systems of polynomial equations with nonnegative coefficients, non-autonomous Hamiltonian systems of Painlevé-type, and polynomial least-squares fitting with noisy or incomplete data (Cartwright, 2010, Dell'Atti et al., 14 Oct 2025).

Given such initial models—e.g., Hamiltonian systems defined on affine or projective spaces, sum-of-squares problems, or systems of orthogonal polynomials—expressions are typically rational in nature and may exhibit problematic behavior (such as 0/0 indeterminacy) at base points or specific parameter values.

The iterative regularisation process targets these issues using algebraic geometry tools (notably, blow-ups and birational transformations), analytic descent algorithms, or information-theoretic updates depending on the context.

2. Blow-ups and Resolution of Indeterminacies

In geometric contexts—prominently for polynomial Hamiltonian systems of Painlevé IV-type or differential systems associated with orthogonal polynomials—the iterative regularisation process centers on systematically removing indeterminacies via coordinate blow-ups (Filipuk et al., 27 Nov 2024, Dell'Atti et al., 14 Oct 2025).

A "blow-up" is a local modification of the ambient space (e.g., ℙ¹ × ℙ¹) to replace an indeterminate point (a, b)—where both numerator and denominator of the defining vector field vanish—by a new space parameterized by charts such as

$q = a + UV, \quad p = b + V$ , or
$q = a + V, \quad p = b + UV$ .

The iterative process repeatedly identifies such base points (singularities) and performs cascades of blow-ups until the vector field becomes regular (i.e., its right-hand side is well-defined) in each chart. Each round of regularisation may expose further singularities requiring additional blow-ups, leading to a tree-like structure of charts and transformations.

The process not only resolves all movable singularities but also decomposes the overall birational transformation between the original, possibly complicated, system and its regularized form into a composition of simpler maps—often Möbius or elementary birational transformations.

3. Newton Polygon Minimality and Selection of Canonical Representatives

Having constructed regularised (polynomial) forms of the dynamical system in each chart, a key principle for identifying a "minimal" system is based on the associated Newton polygon (Dell'Atti et al., 14 Oct 2025).

For a polynomial Hamiltonian $H(x, y; z)$ , the Newton polygon is the convex hull in ℤ² of the exponents (i, j) appearing in the monomials $x^i y^j$ with nonzero coefficients in H.
The iterative process is used not only for normalization but also for selecting the system with minimal Newton polygon area and minimal highest total degree (defined as max $\{ i+j \}$ across (i, j) in the polygon's support).

For instance, in the Painlevé IV context, the Okamoto Hamiltonian provides a minimal Newton polygon (e.g., area 2, highest degree 3), and among all birationally equivalent forms, those satisfying such minimality criteria are taken as canonical (Dell'Atti et al., 14 Oct 2025).

This minimality principle is central to the classification of polynomial Hamiltonian systems and underpins their utility in broader contexts—for example, reductions of partial differential equations and applications to the theory of orthogonal polynomials.

4. Iterative Descent Algorithms for Polynomial Systems

In the context of polynomial systems solved numerically, iterative regularisation is frequently realized through EM-like schemes, gradient-based descent, or iterative proportional fitting (Cartwright, 2010, Després et al., 2018).

For positive solutions to polynomial equations, an expectation-maximization (EM) / iterative proportional fitting (IPF) strategy is employed:

An outer EM-like loop computes weights corresponding to the expected contribution of each monomial.
An inner IPF-like loop updates the parameter vector $x$ via a nonnegative coordinate block update designed to decrease a generalized Kullback–Leibler divergence,
The process is proven to converge to a critical point of the divergence (i.e., an exact non-negative solution if one exists); otherwise, it yields a "regularized" solution minimizing the divergence in the overdetermined or noisy case (Cartwright, 2010).

In the sum-of-squares (SOS) setting, polynomial regularisation is implemented by minimizing a convex functional $G(\lambda) = \operatorname{tr}(M(\lambda)^{-1}) + \langle \lambda, y \rangle$ , with $M(\lambda)$ a moment matrix constructed from the polynomial's structure. Iterative descent methods—including gradient, Newton, and quasi-Newton updates—converge to a unique minimizer, thus recovering an exact or approximate SOS decomposition (Després et al., 2018).

5. Polynomialisation of Differential Systems and Painlevé Equations

The iterative regularisation process is tightly connected to reducing rational dynamical systems to polynomial ones, and further to classical integrable systems such as the Painlevé equations (Filipuk et al., 27 Nov 2024).

After all indeterminacies (including those at infinity) are removed, further local transformations may be applied to achieve polynomial (rather than rational) right-hand sides in the governing differential equations. In practice, this may involve changes such as

$U = -1 + V_2, \quad V = N + U_2 V_2$

leading to systems where all terms in the ODEs or Hamiltonian are polynomial.

These regularised polynomial systems frequently reduce—via elimination or Möbius transformations—to standard forms of Painlevé equations, most notably Painlevé V or Painlevé IV, with parameters explicitly related to the original polynomial system coefficients. For example, a solution variable $U(t)$ in a blown-up chart is mapped by a Möbius transformation $U(t) = -1 + y(t)$ , resulting in a scalar ODE equivalent to $P_V$ (Filipuk et al., 27 Nov 2024).

This iterative reduction exposes the hidden integrable structure of the original polynomial system, enabling explicit parametrisation or classification.

6. Consequences for Birational Geometry and System Classification

By viewing the iterative process as a sequence of birational transformations and blow-ups, the regularisation reveals the geometric underpinnings of complex polynomial systems. The explicit factorisation of the full transformation into simpler components provides a clear geometric picture of the system's moduli, equivalence classes, and singularity structure (Dell'Atti et al., 14 Oct 2025, Filipuk et al., 27 Nov 2024).

Minimality with respect to the Newton polygon not only singles out the simplest representative within a birational equivalence class but also connects algebraic and analytic perspectives—for example, by explaining why certain polynomial Hamiltonians (such as Okamoto's) play a central role in the theory of Painlevé equations and their degenerations.

7. Summary Table: Key Steps in Iterative Polynomial Regularisation

Step	Geometric Context	Analytic/Numeric Context
Identification of singularities	Locate base points in ℙ¹×ℙ¹	Identify problematic equations
Local blow-up/transformation	New charts, exceptional divisors	Coordinate updates, EM/IPF steps
Iterative cascade	Successive blow-ups	Descent iterations, block updates
Regularity and termination	Regular vector field; minimal Newton polygon	Critical point for divergence or G(λ)
Canonicalization	Select minimal area/degree polygon	Select solution minimizing a divergence criterion
Reduction to standard form (e.g., Painlevé)	Möbius or coordinate change	Substitution to scalar ODE

This synthesis highlights the central role of iterative polynomial regularisation as a tool for simplifying, regularising, and uniquely characterising polynomial systems, both in algebraic geometry (via Newton polygons, compactification, and birational geometry) and in numerical analysis (via descent methods, expectation-maximization frameworks, and SOS optimization), establishing a unified structure for tackling nontrivial polynomial dynamical systems and their integrable reductions (Dell'Atti et al., 14 Oct 2025, Filipuk et al., 27 Nov 2024, Cartwright, 2010, Després et al., 2018).