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Existential arithmetization of Diophantine equations

Yuri Matiyasevich
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
The new method leads to a much simpler construction of a universal Diophantine equation and to the existential arithmetization of Turing machines, register machines, and partial recursive functions.  ...  A new method of coding Diophantine equations is introduced.  ...  a new, much simpler purely number-theoretical construction of universal Diophantine equations; • given a code of a Diophantine equation, defining by a purely existential formula the code of an equation  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2008.09.009">doi:10.1016/j.apal.2008.09.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fqokbrm42jcphnu4b2klrs3sqe">fatcat:fqokbrm42jcphnu4b2klrs3sqe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922120430/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/023/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDE2ODAwNzIwODAwMTMyMg%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/32/6a/326a5fd55c9d7a0227d9ac3ac0ae81ee8b90f7d2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2008.09.009"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 437 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 72, Issue 3 [page]

<span title="">1952</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_american-mathematical-society-transactions" style="color: black;">American Mathematical Society. Transactions of the American Mathematical Society </a> </i> &nbsp;
EXISTENTIAL DEFINABILITY IN ARITHMETIC BY JULIA ROBINSON 1. Introduction.  ...  that p(%1,-°+,%n) > Vi Plai-++, Xn, M1, °° +, Me) = 0. ee Thus, a set of natural numbers is existentially definable if and only if it is the set of values of a parameter for which a certain diophantine  ... 
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Approaching Arithmetic Theories with Finite-State Automata [chapter]

Christoph Haase
<span title="">2020</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
This approach has recently been instrumental for settling long-standing open problems about the complexity of deciding the existential fragments of Büchi arithmetic and linear arithmetic over p-adic fields  ...  In this article, which accompanies an invited talk, we give a high-level exposition of the NP upper bound for existential Büchi arithmetic, obtain some derived results, and further discuss some open problems  ...  reachability queries in p-automata implicitly given by systems of linear Diophantine equations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-030-40608-0_3">doi:10.1007/978-3-030-40608-0_3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y4yvocbpvzdcba4jlsttgv2nwq">fatcat:y4yvocbpvzdcba4jlsttgv2nwq</a> </span>
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On the Expressiveness of Büchi Arithmetic [chapter]

Christoph Haase, Jakub Różycki
<span title="">2021</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Furthermore, we show that regular languages of polynomial growth are definable in the existential fragment of Büchi arithmetic.  ...  AbstractWe show that the existential fragment of Büchi arithmetic is strictly less expressive than full Büchi arithmetic of any base, and moreover establish that its $$\varSigma _2$$ Σ 2 -fragment is already  ...  We would like to thank Dmitry Chistikov and Alex Fung for inspiring discussions on the topics of this paper, and the FoSSaCS'21 reviewers for their comments and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-030-71995-1_16">doi:10.1007/978-3-030-71995-1_16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x3fbuyuspnaonec6x6x54wptv4">fatcat:x3fbuyuspnaonec6x6x54wptv4</a> </span>
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On the Expressiveness of Büchi Arithmetic [article]

Christoph Haase, Jakub Różycki
<span title="2021-03-02">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Furthermore, we show that regular languages of polynomial growth are definable in the existential fragment of Büchi arithmetic.  ...  We show that the existential fragment of Büchi arithmetic is strictly less expressive than full Büchi arithmetic of any base, and moreover establish that its Σ_2-fragment is already expressively complete  ...  We would like to thank Dmitry Chistikov and Alex Fung for inspiring discussions on the topics of this paper, and the FoSSaCS'21 reviewers for their comments and suggestions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2010.12892v3">arXiv:2010.12892v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pffgsaaxhfg6xargsgtlopnmra">fatcat:pffgsaaxhfg6xargsgtlopnmra</a> </span>
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On the Existential Theories of Büchi Arithmetic and Linear p-adic Fields

Florent Guepin, Christoph Haase, James Worrell
<span title="">2019</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cnybbxuptncftdgxtodn5edz7m" style="color: black;">2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)</a> </i> &nbsp;
We consider the complexity of the satisfiability problems for the existential fragment of Büchi arithmetic and for the existential fragment of linear arithmetic over p-adic fields.  ...  A key technical contribution is to show that the existence of a path between two states of a finitestate automaton whose language encodes the set of solutions of a given system of linear Diophantine equations  ...  Without loss of generality, we can assume that atomic formulas of Büchi arithmetic are either linear Diophantine equations a·x = c or assertions V p (x) = y.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/lics.2019.8785681">doi:10.1109/lics.2019.8785681</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/lics/GuepinH019.html">dblp:conf/lics/GuepinH019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j6zijkzgenhtjot7bkyrtbtc4e">fatcat:j6zijkzgenhtjot7bkyrtbtc4e</a> </span>
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Existential definability in arithmetic

Julia Robinson
<span title="1952-03-01">1952</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
deciding if a given diophantine equation is solvable.  ...  At present, very little is known about the size of solutions of diophantine equations with a finite number of solutions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-1952-0048374-2">doi:10.1090/s0002-9947-1952-0048374-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ya5bag3slfhy5iy6a4ky5gkthm">fatcat:ya5bag3slfhy5iy6a4ky5gkthm</a> </span>
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Existential Definability in Arithmetic

Julia Robinson
<span title="">1952</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
deciding if a given diophantine equation is solvable.  ...  At present, very little is known about the size of solutions of diophantine equations with a finite number of solutions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/1990711">doi:10.2307/1990711</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d3kpxliygfh7rcsihdzobiltva">fatcat:d3kpxliygfh7rcsihdzobiltva</a> </span>
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Two conjectures on integer arithmetic and their applications to Diophantine equations [article]

Apoloniusz Tyszka
<span title="2010-12-01">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let T be a recursive axiomatization of arithmetic. In the above proof, we showed the existence of a Diophantine equation that is undecidable in T ∪{Conjecture 3}, if this theory is consistent.  ...  Of course, each set B n (Z) is listable as an existentially definable subset of Z n . Theorem 3 illustrates how strong Conjecture 3 is.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0911.0384v58">arXiv:0911.0384v58</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ggojeagmg5cg3kzs6uwqy4qr4q">fatcat:ggojeagmg5cg3kzs6uwqy4qr4q</a> </span>
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Decidable Sentences Over Polynomial Rings

Shih-Ping Tung
<span title="">1988</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
Our main tools are theorems of Schinzel [9] on diophantine equations with parameters.  ...  Problems of this type are called diophantine equations with parameters in number theory.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2045893">doi:10.2307/2045893</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7peflqpytfbgjf753cdl62r7da">fatcat:7peflqpytfbgjf753cdl62r7da</a> </span>
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Decidable sentences over polynomial rings

Shih Ping Tung
<span title="1988-02-01">1988</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
Our main tools are theorems of Schinzel [9] on diophantine equations with parameters.  ...  Problems of this type are called diophantine equations with parameters in number theory.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-1988-0921004-0">doi:10.1090/s0002-9939-1988-0921004-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wfvqt5rg6rhfxmpisrtgqvkznu">fatcat:wfvqt5rg6rhfxmpisrtgqvkznu</a> </span>
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Hilbert's Tenth Problem

Alexandra Shlapentokh
<span title="2021-04-01">2021</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/o5hwtvfzwza2tdpaa2xm3sqpqe" style="color: black;">Notices of the American Mathematical Society</a> </i> &nbsp;
These connections between Diophantine definability and some of the most interesting questions in mathematics tell us that the existential language of rings, i.e., the existential language of polynomial  ...  A Diophantine definition of the exponential function obtained via the Pell equation is discussed in Section 3 of the chapter.  ...  The main goal of the chapter is to explain the statement of a theorem proved by Ram Murty and Hector Pasten [MP2018] connecting Hilbert's Tenth Problem for rings of integers of number fields to several  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/noti2249">doi:10.1090/noti2249</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qxgloxnxafcwrev62efmmjqolm">fatcat:qxgloxnxafcwrev62efmmjqolm</a> </span>
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Undecidability in number theory [article]

Jochen Koenigsmann
<span title="2013-09-02">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
It also contains a sketch of the authors result that the integers are universally definable in the rationals.  ...  equations, asked for a more modest algorithm that decides whether or not a given diophantine equation has a solution.  ...  In the 3rd century AD, Diophantos of Alexandria, often considered the greatest (if not only) algebraist of antique times, tackled what we call today diophantine equations, that is, polynomial equations  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1309.0441v1">arXiv:1309.0441v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ndb52ta63rdilgojjewavp2wqi">fatcat:ndb52ta63rdilgojjewavp2wqi</a> </span>
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Topology of Diophantine Sets: Remarks on Mazur's Conjectures [article]

Gunther Cornelissen, Karim Zahidi
<span title="2000-06-20">2000</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers.  ...  We also prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over that finite field. Both proofs depend upon Matijasevich's theorem.  ...  (d) To give an example with a different language, existentially definable sets of Z in the language (0, 1, +, |) are unions of arithmetic progressions (a result of Lipshitz [11] ).  ... 
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Page 8591 of Mathematical Reviews Vol. , Issue 2001M [page]

<span title="">2001</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
It asks for an algorithm which, given an arbitrary Diophantine equation, checks whether the equation has integer solutions.  ...  The second problem considers the existential theory of the field of rational numbers. It is known that the existential theory of many rational func- tion fields is undecidable.  ... 
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