Optimal Lojasiewicz-Simon Exponents and the Intrinsic Singularities of Analytic Functionals
Abstract
We prove that the optimal Lojasiewicz--Simon exponent of a real-analytic functional on a Banach space is an intrinsic invariant of its finite-dimensional Lyapunov--Schmidt singularity under natural Fredholm and pairing-compatible hypotheses. More precisely, Lyapunov--Schmidt reduction preserves the entire set of admissible exponents, while the reduced germ on the kernel of the Hessian is intrinsic up to analytic right-equivalence tangent to the identity. We then identify this invariant with a real relative Lojasiewicz exponent of the pair consisting of the Jacobian ideal of the reduced energy and its principal energy ideal. Consequently, the optimal exponent admits equivalent descriptions by analytic arcs and by a finite maximum of divisorial ratios; in particular, it is rational. In Newton-nondegenerate cases, these formulas yield explicit monomial expressions. We apply the theory to several classes of singular energies. For the sum Qk of the squares of all k× k minors of a real matrix, we determine the associated ideal pair up to real integral closure and obtain the exact local exponent 1−1/(2(k−s)) at matrices of rank s<k. For Yang--Mills energy, we prove that the product flat connection on 𝕋2 has optimal exponent 3/4 for every nonabelian compact structure group. For SU(2), the determinantal reduction extends this value to product flat connections on every 𝕋d, while sufficiently small nonzero constant flat connections have exponent 1/2. Finally, for resonant semilinear Dirichlet energies, we identify higher-order resonant obstructions, establish an all-order obstruction ladder for simple resonance, and obtain a complete parity-dependent classification on (0,π).
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BibTeX
@misc{amdeberhan2026optimal,
title = {{Optimal Lojasiewicz-Simon Exponents and the Intrinsic Singularities of Analytic Functionals}},
author = {Tewodros Amdeberhan and Tai Huy Ha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29918},
eprint = {2609.29918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29918},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Lojasiewicz-Simon Exponents and the Intrinsic Singularities of Analytic Functionals
AU - Tewodros Amdeberhan
AU - Tai Huy Ha
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29918
UR - https://arxiv.org/abs/2609.29918
AB - We prove that the optimal Lojasiewicz--Simon exponent of a real-analytic functional on a Banach space is an intrinsic invariant of its finite-dimensional Lyapunov--Schmidt singularity under natural Fredholm and pairing-compatible hypotheses. More precisely, Lyapunov--Schmidt reduction preserves the entire set of admissible exponents, while the reduced germ on the kernel of the Hessian is intrinsic up to analytic right-equivalence tangent to the identity. We then identify this invariant with a real relative Lojasiewicz exponent of the pair consisting of the Jacobian ideal of the reduced energy and its principal energy ideal. Consequently, the optimal exponent admits equivalent descriptions by analytic arcs and by a finite maximum of divisorial ratios; in particular, it is rational. In Newton-nondegenerate cases, these formulas yield explicit monomial expressions. We apply the theory to several classes of singular energies. For the sum Q_k of the squares of all k× k minors of a real matrix, we determine the associated ideal pair up to real integral closure and obtain the exact local exponent 1−1/(2(k−s)) at matrices of rank s<k. For Yang--Mills energy, we prove that the product flat connection on 𝕋^2 has optimal exponent 3/4 for every nonabelian compact structure group. For SU(2), the determinantal reduction extends this value to product flat connections on every 𝕋^d, while sufficiently small nonzero constant flat connections have exponent 1/2. Finally, for resonant semilinear Dirichlet energies, we identify higher-order resonant obstructions, establish an all-order obstruction ladder for simple resonance, and obtain a complete parity-dependent classification on (0,π).
ER -