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 $Q_k$ of the squares of all $k\times k$ minors of a real matrix, we determine the associated ideal pair up to real integral closure and obtain the exact local exponent $$1-\frac1{2(k-s)}$$ at matrices of rank $s<k$. For Yang--Mills energy, we prove that the product flat connection on $\mathbb T^2$ has optimal exponent $3/4$ for every nonabelian compact structure group. For $\mathrm{SU}(2)$, the determinantal reduction extends this value to product flat connections on every $\mathbb T^d$, 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 $(0,π)$.