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{
    "code": 0,
    "data": {
        "papers": [
            {
                "paper": {
                    "package_id": "paper:1000399181526859777",
                    "id": "1000399181526859777",
                    "doi": "10.48550/arxiv.2405.12771",
                    "publication_id": "108545",
                    "publication_name": "Logic _ Mathematics.math.LO",
                    "zh_title": "域的全称-存在性理论",
                    "en_title": "Universal-existential theories of fields",
                    "authors": "Sylvy Anscombe | Arno Fehm",
                    "publication_date": "2024-05-21",
                    "available_online": "",
                    "cover_date_start": "2024-05-21",
                    "area": "arXiv-Mathematics",
                    "research_categories": "[\"Mathematical Sciences\", \"Pure Mathematics\"]",
                    "keywords": "",
                    "created_at": "2024-05-23T05:53:19+08:00"
                },
                "addressed_problems": [
                    {
                        "id": "paper:1000399181526859777::problem",
                        "global_id": "gcn_cc4a057cc82b4bc2",
                        "content": "The paper addresses the following pre-contribution problem-state. Model-theoretic and computability questions about fields, their rational function fields $k(t)$, function fields $k(X)$, and equicharacteristic henselian valued fields (in particular Laurent series fields $k((t))$) are only partially understood when attention is restricted to syntactic fragments between purely existential sentences and full first-order theories. Classical results (Ax–Kochen–Ershov) give transfer principles and decidability for full theories in characteristic zero valued fields, and several works established many-one reductions and decidability-lifting for pure existential fragments (e.g. $\\mathrm{Th}_{\\exists}(k)$ to $\\mathrm{Th}_{\\exists}(k((t)),v_t)$). However, for universal-existential fragments (sentences of the form $\\forall\\exists$ or variants with bounded numbers of universal quantifiers, or universal quantifiers restricted to the residue field), the following limitations remained in the literature and motivated this work:\n    \n    - Lack of systematic reductions: There was no systematic set of computable many-one reductions relating $\\mathrm{Th}_{\\forall\\exists}$ and the bounded-quantifier fragments $\\mathrm{Th}_{\\forall_n\\exists}$ (including $\\mathrm{Th}_{\\forall_k\\exists}$ where universals range only over the residue field) between a field $k$ (or its residue field) and the corresponding theories of $k(t)$, $k(X)$ or $k((t))$ with or without parameters (such as the constant symbol $t$). Existing existential reductions do not directly give such reductions for universal-existential fragments because universal quantifiers are harder to eliminate or reduce.\n    \n    - Positive characteristic obstructions: In characteristic $p>0$ the presence of purely inseparable extensions and feature of $p$-powers (e.g. $[K:K^p]=p^n$ and $p$-bases) creates the possibility of encoding several universal quantifiers by fewer universals plus existential data, but no uniform, computable formulation of this \"quantifier-coding\" using $p$-th powers had been provided for fragments like $\\forall\\exists$ or $\\forall_n\\exists$.\n    \n    - Valued-field fragments: For equicharacteristic henselian valued fields, although the existential theory of the residue field controls existential sentences in many cases, it was unclear to what extent $\\mathrm{Th}_{\\forall\\exists}$ (or residue-field-only universal quantifiers $\\forall_{\\mathfrak{k}}\\exists$) of the residue field determines the corresponding universal-existential fragments of the valued field, or whether computable many-one reductions could be given. The classical Ax–Kochen/Ershov principles address full theories in characteristic zero but do not directly yield reductions for these restricted fragments, and positive-characteristic analogues often rely on conditional geometric hypotheses (e.g. (R4)).\n    \n    - Decidability consequences and intermediate fragments: There was no comprehensive transfer of decidability (and many-one equivalences) between various fragments (e.g. $\\mathrm{Th}_{\\exists}(k(t),t)$, $\\mathrm{Th}_{\\forall_1\\exists}(k(t))$, $\\mathrm{Th}_{\\forall\\exists}(k(t))$, and their valued-field analogues) under general hypotheses on $k$ (such as being perfect, large, finite, or of specified characteristic). Consequentially, undecidability results for existential fragments were not systematically lifted to universal-existential fragments.\n    \n    - Concrete negative obstructions: It was not settled whether simple existential definability obstructions prevent certain naive many-one reductions (for instance, whether there is an $\\exists$-definable surjection $k((t))\\to k((t))\\times k((t))$ over perfect $k$).\n    \n    These gaps meant that for many natural fiel