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Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

Assia Mahboubi, Cyril Cohen, Henk Barendregt
2012 Logical Methods in Computer Science  
After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination  ...  This paper describes a formalization of discrete real closed fields in the Coq proof assistant.  ...  Acknowledgments The authors wish to thank Georges Gonthier for his precious suggestion to use continuationpassing style in the last part of this work.  ... 
doi:10.2168/lmcs-8(1:2)2012 fatcat:kespb2lx6rh6dl3ykz65ufjfcu

Real algebraic geometry

2014 ChoiceReviews  
the pool of the Real Algebraic Geometry of tomorrow.  ...  We also wish to thank the numerous sponsors who allowed us to welcome so many participants under proper conditions, in particular a large number of PhD students and post-PhD students called to constitute  ...  Introduction We survey developments in the theory of algorithms in real algebraic geometry -starting from the first effective quantifier elimination procedure due to Tarski and Seidenberg, to more recent  ... 
doi:10.5860/choice.51-3296 fatcat:xm5l6ieafzewtfpuegkxsbtzhy

Some results from algebraic geometry over complete discretely valued fields [article]

Krzysztof Jan Nowak
2016 arXiv   pre-print
Our approach applies the quantifier elimination due to Pas. By the transfer principle of Ax-Kochen-Ershov, all these results carry over to the case of Henselian discretely valued fields.  ...  This paper is concerned with algebraic geometry over complete discretely valued fields K of equicharacteristic zero.  ...  Proof of Theorem 1. We begin with quantifier elimination due to Pas for Henselian valued fields K with residue field k of characteristic zero.  ... 
arXiv:1311.2051v5 fatcat:4kanrydbajegrfeklb6ofpihmi

Proof and Computation in Geometry [chapter]

Michael Beeson
2013 Lecture Notes in Computer Science  
We might try to produce such proofs directly, or we might try to develop a "backtranslation" from algebra to geometry, following Descartes but with computer in hand.  ...  But this does not produce computer-checkable first-order proofs in geometry.  ...  Similarly every model of Tarski geometry is F 2 , where F is real-closed. The smallest model of Tarski geometry corresponds to the case when F is the field of real algebraic numbers.  ... 
doi:10.1007/978-3-642-40672-0_1 fatcat:syvmbmvp7je7rpppujtew7jo7a

Some results of algebraic geometry over Henselian rank one valued fields

Krzysztof Jan Nowak
2016 Selecta Mathematica, New Series  
We develop geometry of affine algebraic varieties in K^n over Henselian rank one valued fields K of equicharacteristic zero.  ...  Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection.  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in  ... 
doi:10.1007/s00029-016-0245-y fatcat:mkxxnlu66jbcjpgsltqtatka5a

CONSTRUCTIVE GEOMETRY

MICHAEL BEESON
2009 Proceedings of the 10th Asian Logic Conference  
ECG is axiomatized in a quantifier-free, disjunction-free way. Unlike previous intuitionistic geometries, it does not have apartness.  ...  We also study the formal relationships between several versions of Euclid's parallel postulate, and show that each corresponds to a natural axiom system for Euclidean fields. 1  ...  Projection is absolutely necessary in order to reduce geometry to algebra.  ... 
doi:10.1142/9789814293020_0002 fatcat:gxbw3oexhbcmpg2qzpzubb4upm

Can one design a geometry engine? On the (un)decidability of affine Euclidean geometries [article]

J.A. Makowsky
2018 arXiv   pre-print
We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski's conjecture on the undecidability of finitely axiomatizable theories of fields.  ...  We survey the status of decidabilty of the consequence relation in various axiomatizations of Euclidean geometry.  ...  Bernays used to live in Göttingen, before going into forced exile in 1933. P. Bernays edited Hilbert's [Hil02] from the 5th (1922) till the 10th edition (1967), see also [Hil71, Hil13].  ... 
arXiv:1712.07474v3 fatcat:tpd4slz6ergmpetkwcx5ex25pq

Combinational Convexity and Algebraic Geometry

1997 Journal of Computational and Applied Mathematics  
Linear algebra -solving linear systems. The method of Gro'bner bases. Quantifier elimination in real closed fields. 10. Indefinite summation. Parametrization of algebraic curves.  ...  In the second part, Bernshtein's theorem is proved using proper concepts from algebraic geometry, so that we can indeed consider mixed volumes as intersection numbers.  ... 
doi:10.1016/s0377-0427(97)81613-5 fatcat:4xekr6leprdudaecydsf3qju24

Invariant computations for analytic projective geometry

Walter Whiteley
1991 Journal of symbolic computation  
From a proof of an open theorem about "geometric properties", over all fields, or over ordered fields, an algorithm derives Nullstellensatz identities -giving maximal algebraic simplicity, and maximal  ...  This special form corresponds to statements in synthetic projective geometry and the algorithm is a basic step towards translation back into synthetic geometry.  ...  In Section 10, we present new analogues of these results for projective geometry over the reals (i.e. ordered fields and real-closed fields).  ... 
doi:10.1016/s0747-7171(08)80119-8 fatcat:3gzloxwkpzexve2wxqknjwvoyi

Some results from algebraic geometry over Henselian real valued fields [article]

Krzysztof Jan Nowak
2016 arXiv   pre-print
This paper develops algebraic geometry over Henselian real valued (i.e. of rank 1) fields K, being a sequel to our paper about that over Henselian discretely valued fields.  ...  Our approach applies quantifier elimination due to Pas.  ...  We endeavour to carry over the results from algebraic geometry over Henselian discretely valued fields of equicharacteristic zero, given in our paper [21] , to that over Henselian real valued (i.e. of  ... 
arXiv:1312.2935v6 fatcat:76vcm44gb5b65pj22ooo5bvnwi

Autonomy of Geometry

John T. Baldwin, Andreas Mueller
2020 Annales Universitates Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia  
project to axiomatize Euclid's geometry in a first order geometric language, notably eliminating the dependence on the Archimedean axiom; (3) the independent conception of multiplication from a geometric  ...  In this paper we present three aspects of the autonomy of geometry. (1) An argument for the geometric as opposed to the 'geometric algebraic' interpretation of Euclid's Books I and II; (2) Hilbert's successful  ...  Baldwin, Andreas Mueller The key to Hilbert's elimination of the axiom of Achimedes is to define a field from first order geometric principles.  ... 
doi:10.24917/20809751.11.1 fatcat:44mlmr54arbbpfmqqapbdliod4

A closedness theorem and applications in geometry of rational points over Henselian valued fields [article]

Krzysztof Jan Nowak
2018 arXiv   pre-print
Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary  ...  This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields.  ...  Two basic tools applied in this paper are quantifier elimination for Henselian valued fields (along with preparation cell decomposition) due to Pas [51] and relative quantifier elimination for ordered  ... 
arXiv:1706.01774v6 fatcat:3iceggz5lrhmhcjshb6ekfgkpi

Algorithms in Real Algebraic Geometry: A Survey [article]

Saugata Basu
2014 arXiv   pre-print
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg  ...  We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic  ...  Tarski's proof [87] of the existence of quantifier elimination in the first order theory of the reals is effective and is based on Sturm's theorem for counting real roots of polynomials in one variable  ... 
arXiv:1409.1534v1 fatcat:nyprfglktvdtnmhu3zwqrb547y

Book Review: Zariski geometries. Geometry from the logician's point of view

Anand Pillay
2012 Bulletin of the American Mathematical Society  
from analytic rather than algebraic geometry.  ...  n with a family Z n of distinguished definable subsets of M n , which we will call the closed or definable closed sets in M , with the following properties: (i) Quantifier elimination to closed sets.  ... 
doi:10.1090/s0273-0979-2012-01373-3 fatcat:5fmhs6gsw5cpfjaqjid666z76u

Real Algebraic Geometry and Constraint Databases [chapter]

Floris Geerts, Bart Kuijpers
2007 Handbook of Spatial Logics  
The constraint database model can be seen as a generalization of the classical relational database model that was introduced by Codd in the 1970s to deal with the management of alpha-numerical data, typically  ...  in business applications (?).  ...  However, since the triviality theorem does not hold in this case, one first needs to develop an alternative decomposition theorem (see (?) for details).  ... 
doi:10.1007/978-1-4020-5587-4_13 fatcat:5oemzcnwe5egnkkrjxquwsl5ve
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