Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.21738

Local definable $(p,q)$-theorem and a refinement

Gareth John Boxall·Charlotte Kestner·Tsinjo Rakotonarivo
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove the definable $(p,q)$-conjecture for NIP formulas in arbitrary first-order theories. We also prove a refinement and apply it to dense pairs.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gareth John Boxall, Charlotte Kestner, Tsinjo Rakotonarivo. 2026-10-15. Local definable $(p,q)$-theorem and a refinement. https://arxiv.org/abs/2610.21738

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extended Frege proofs, circuits and rewriting

Inspired by a statement about Extended Frege proof systems by Jain and Jin (FOCS 2022) we prove that: - there is a p-time binary relation $\approx$ between circuits that implies their logical equivalence, - if the equivalence $C \equiv D$ has an Extended Frege or a Circuit Frege proof with $k$ steps then there is a chain of circuits $E_i$ $$ C = E_0 \approx \dots \approx E_t = D $$ with $t \le O(k \log k)$, and the total size of the chain is bounded above by $(k+|C|+|D|)^{O(1)}$.

math.LO↗
arXiv2026

Canonical expansions of $\mathbb R_{\mathcal Q}$-germs

Let $\mathcal{H}_{\mathcal{Q}}$ be the field of unary germs at $0^+$ definable in the quasianalytic o-minimal structure $\mathbb{R}_{\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of $\mathcal{H}_{\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that $\mathbb{R}_{\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic $\mathbb{R}_{\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary $\mathbb{R}_{\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of $\mathcal{H}_{\mathcal{Q}}$.

math.LO↗
arXiv2026

Discrete Temporal CSPs et al

We revisit the complexity classification of constraint satisfaction problems (CSPs) for first-order reducts of $(\mathbb{Z},<)$, established by Bodirsky, Martin, and Mottet in 2018. Following their approach, we give a self-contained proof of this classification, correcting several errors in the argument. We further show that, with the exception of cases arising from finite-domain CSPs, all tractable cases are solved by a well-known uniform algorithm, eliminating the need for the collection of ad-hoc algorithms used in the original paper. A central component of the proof allows the analysis to be restricted to first-order reducts of $(\mathbb{Z},<)$ that contain the successor relation. We substantially simplify this reduction and generalize the underlying construction. This generalization allows us to apply the same technique to first-order reducts of the triangular lattice graph and of infinite regular trees, yielding a complete complexity classification for these classes of structures, in particular a $\mathsf{P}$ versus $\mathsf{NP}$-complete dichotomy.

math.LO↗