{"ID":23620897,"CreatedAt":"2026-09-18T06:47:37.825099056Z","UpdatedAt":"2026-09-18T06:47:37.825099056Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20668","arxiv_id":"2609.20668","title":"Logarithmic--exponential preparation in sharply o-minimal structures","abstract":"We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\\mathbb R}^\\text{RE}_{\\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion--Rolin. We also develop a parallel complex theory for ${\\mathbb R}_\\text{an,exp}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\\text{poly}(\\log H)$. This generalizes a theorem of Cluckers--Pila--Wilkie who proved the same statement for ${\\mathbb R}_\\text{an}^\\text{pow}$.","short_abstract":"We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\\mathbb R}^\\text{RE}_{\\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wil...","url_abs":"https://arxiv.org/abs/2609.20668","url_pdf":"https://arxiv.org/pdf/2609.20668v1","authors":"[\"Gal Binyamini\",\"Oded Carmon\",\"Dmitry Novikov\"]","published":"2026-09-17T16:43:27Z","proceeding":"math.LO","tasks":"[\"math.LO\",\"math.AG\",\"math.NT\"]","methods":"[]","has_code":false}