The shared-eigenvector idea
A preprint presents a symbolic-numeric eigensolver for algebraic problems, built around joint eigenvectors of commuting operators. Its stated aim is to recover roots, analyze how often roots occur with multiplicity, and implement the approach in AlgebraicSolvers.jl. The work is a methods paper with a polynomial-system example and a symmetric-tensor example.
The central theorem gives the method its bridge from algebra to linear algebra: common eigenvectors of transposed multiplication operators are, up to scale, evaluation functionals at the roots. In ordinary language, the operators are set up so that vectors shared by them carry information about where the solutions are. That shared structure is what lets the calculation address repeated solutions as well as distinct locations.
To do the computation, the workflow first computes a truncated normal form, or TNF, and extracts a basis. It then builds multiplication matrices, forms a random matrix combination to obtain eigenvalues, and chooses either joint diagonalization or joint Schur factorization before reporting roots and multiplicities. Multiplicity classification uses a clustering function with a default threshold of 0.00001, meaning numerical values close enough to be grouped are treated as belonging together.
What the examples returned
The first illustration is a polynomial system made up of two equations in two variables. Using joint Schur factorization, it produced solution columns approximately at [0, 0] and [-2, 2]. The reported multiplicities were 3 for the first solution and 1 for the second, so the calculation treated the point at the origin as a repeated solution.
The authors also compare two ways to build the TNF. In this example, the resultant or Macaulay route produced the same basis as the Gröbner-basis route and the same order of numerical root accuracy. That is a result about agreement in one illustrated system, not a general ranking of the two constructions.
A tensor version of the method
The second illustration shifts from polynomial systems to tensor decomposition. It treats a symmetric tensor as, equivalently, a homogeneous polynomial or form, then represents that form as a sum of terms in which a weight polynomial is multiplied by a power of a linear form. This generalized additive decomposition is the format to which the eigensolver machinery is applied.
For the tensor example, the reported point-columns encoding the linear forms were [1, -2, 2] and [1, 0, 0]. The degrees of the two homogeneous weights were reported as (0, 2), obtained from the nil-indices of local multiplication operators. In practical terms, the example paired one degree-zero weight with one degree-two weight.
The calculation also examined the repeated point at [0, 0]. It displayed an inverse-system basis and stated that the corresponding local-algebra basis is dual to it. That output describes more than the location of the point: it records part of the algebraic structure associated with the multiple solution.
Useful evidence, limited reach
The reconstructed tensor had a reported apolar_norm error of 9.342121376789858e-14. That number is a diagnostic from the single reconstruction example; the supplied analysis does not treat it as an error bound or a guarantee that the procedure will converge in other cases.
The examples show the kinds of outputs the workflow can produce, but they do not amount to a broad performance evaluation. No systematic benchmark, runtime analysis, distribution of accuracy, replication study, or convergence evaluation is reported. The Gröbner-versus-Macaulay comparison is qualitative and limited to one polynomial example, while the effects of the clustering threshold and other numerical choices were not tested systematically. The evidence therefore cannot establish that AlgebraicSolvers.jl is faster, more accurate, or more stable than competing methods.
The document is an arXiv version 2 preprint dated 14 Sep 2026. It describes the implementation in AlgebraicSolvers.jl, and its acknowledgment reports support from the European Union's Horizon Europe Marie Skłodowska-Curie Actions under grant agreement 101120296, known in the acknowledgment as TENORS.
Paper data and sources
Original title: Eigensolvers for polynomial roots and tensor decomposition
Authors: Enrica Barrilli, Bernard Mourrain
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text