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

2012
*
Logical Methods in Computer Science
*

After defining an abstract structure of discrete

doi:10.2168/lmcs-8(1:2)2012
fatcat:kespb2lx6rh6dl3ykz65ufjfcu
*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. ...##
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Real algebraic geometry

2014
*
ChoiceReviews
*

the pool of the

doi:10.5860/choice.51-3296
fatcat:xm5l6ieafzewtfpuegkxsbtzhy
*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 ...##
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Some results from algebraic geometry over complete discretely valued fields
[article]

2016
*
arXiv
*
pre-print

Our approach applies the

arXiv:1311.2051v5
fatcat:4kanrydbajegrfeklb6ofpihmi
*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. ...##
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Proof and Computation in Geometry
[chapter]

2013
*
Lecture Notes in Computer Science
*

We might try

doi:10.1007/978-3-642-40672-0_1
fatcat:syvmbmvp7je7rpppujtew7jo7a
*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. ...##
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Some results of algebraic geometry over Henselian rank one valued fields

2016
*
Selecta Mathematica, New Series
*

We develop

doi:10.1007/s00029-016-0245-y
fatcat:mkxxnlu66jbcjpgsltqtatka5a
*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*...##
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CONSTRUCTIVE GEOMETRY

2009
*
Proceedings of the 10th Asian Logic Conference
*

ECG is axiomatized

doi:10.1142/9789814293020_0002
fatcat:gxbw3oexhbcmpg2qzpzubb4upm
*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*. ...##
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Can one design a geometry engine? On the (un)decidability of affine Euclidean geometries
[article]

2018
*
arXiv
*
pre-print

We draw attention

arXiv:1712.07474v3
fatcat:tpd4slz6ergmpetkwcx5ex25pq
*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]. ...##
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Combinational Convexity and Algebraic Geometry

1997
*
Journal of Computational and Applied Mathematics
*

Linear

doi:10.1016/s0377-0427(97)81613-5
fatcat:4xekr6leprdudaecydsf3qju24
*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. ...##
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Invariant computations for analytic projective geometry

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*). ...

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

2016
*
arXiv
*
pre-print

This paper develops

arXiv:1312.2935v6
fatcat:76vcm44gb5b65pj22ooo5bvnwi
*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 ...##
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Autonomy of Geometry

2020
*
Annales Universitates Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
*

project

doi:10.24917/20809751.11.1
fatcat:44mlmr54arbbpfmqqapbdliod4
*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. ...##
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A closedness theorem and applications in geometry of rational points over Henselian valued fields
[article]

2018
*
arXiv
*
pre-print

Two basic tools applied

arXiv:1706.01774v6
fatcat:3iceggz5lrhmhcjshb6ekfgkpi
*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*...##
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Algorithms in Real Algebraic Geometry: A Survey
[article]

2014
*
arXiv
*
pre-print

We survey both old and new developments

arXiv:1409.1534v1
fatcat:nyprfglktvdtnmhu3zwqrb547y
*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 ...##
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Book Review: Zariski geometries. Geometry from the logician's point of view

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. ...

##
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Real Algebraic Geometry and Constraint Databases
[chapter]

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

doi:10.1007/978-1-4020-5587-4_13
fatcat:5oemzcnwe5egnkkrjxquwsl5ve
*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). ...
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