Resultant multiplicity via projective degrees and applications to tensor eigenvalues
Abstract
Given a system f=(f1,…,fn) of n homogeneous forms in n variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at f. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by f. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by f. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.
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BibTeX
@misc{dogan2026resultant,
title = {{Resultant multiplicity via projective degrees and applications to tensor eigenvalues}},
author = {Mahmut Levent Doğan and Elias Tsigaridas and Zafeirakis Zafeirakopoulos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01268},
eprint = {2609.01268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01268},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Resultant multiplicity via projective degrees and applications to tensor eigenvalues
AU - Mahmut Levent Doğan
AU - Elias Tsigaridas
AU - Zafeirakis Zafeirakopoulos
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01268
UR - https://arxiv.org/abs/2609.01268
AB - Given a system f=(f_1,…,f_n) of n homogeneous forms in n variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at f. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by f. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by f. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.
ER -