Algebraic geometry: Difference between revisions
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* [[#Computational algebraic geometry|Computational algebraic geometry]] is an area that has emerged at the intersection of algebraic geometry and [[computer algebra]], with the rise of computers. It consists mainly of [[algorithm]] design and [[software]] development for the study of properties of explicitly given algebraic varieties. | * [[#Computational algebraic geometry|Computational algebraic geometry]] is an area that has emerged at the intersection of algebraic geometry and [[computer algebra]], with the rise of computers. It consists mainly of [[algorithm]] design and [[software]] development for the study of properties of explicitly given algebraic varieties. | ||
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, [[differential geometry|differential]] and [[complex geometry]]. One key achievement of this abstract algebraic geometry is [[Grothendieck]]'s [[scheme theory]] which allows one to use [[sheaf theory]] to study algebraic varieties in a way which is very similar to its use in the study of [[differential manifold|differential]] and [[complex manifold|analytic manifolds]]. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through [[Hilbert's Nullstellensatz]], with a [[maximal ideal]] of the [[coordinate ring]], while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a | Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, [[differential geometry|differential]] and [[complex geometry]]. One key achievement of this abstract algebraic geometry is [[Grothendieck]]'s [[scheme theory]] which allows one to use [[sheaf theory]] to study algebraic varieties in a way which is very similar to its use in the study of [[differential manifold|differential]] and [[complex manifold|analytic manifolds]]. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through [[Hilbert's Nullstellensatz]], with a [[maximal ideal]] of the [[coordinate ring]], while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a sub-variety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. [[Wiles's proof of Fermat's Last Theorem|Wiles' proof]] of the longstanding conjecture called [[Fermat's Last Theorem]] is an example of the power of this approach. | ||
==Basic notions== | ==Basic notions== | ||
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=== Morphism of affine varieties === | === Morphism of affine varieties === | ||
Using regular functions from an affine variety to '''A'''<sup>1</sup>, we can define [[morphism of algebraic varieties|regular map]]s from one affine variety to another. First we will define a regular map from a variety into affine space: Let ''V'' be a variety contained in '''A'''<sup>''n''</sup>. Choose ''m'' regular functions on ''V'', and call them ''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>. We define a ''regular map'' ''f'' from ''V'' to '''A'''<sup>''m''</sup> by letting {{nowrap|1=''f'' = (''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>)}}. In other words, each ''f''<sub>''i''</sub> determines one coordinate of the [[image (mathematics)|range]] of ''f''. | Using regular functions from an affine variety to '''A'''<sup>1</sup>, we can define [[morphism of algebraic varieties|regular map]]s from one affine variety to another. First we will define a regular map from a variety into affine space: Let ''V'' be a variety contained in '''A'''<sup>''n''</sup>. Choose ''m'' regular functions on ''V'', and call them ''f''<sub>1</sub>, . . ., ''f''<sub>''m''</sub>. We define a ''regular map'' ''f'' from ''V'' to '''A'''<sup>''m''</sup> by letting {{nowrap|1=''f'' = (''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>)}}. In other words, each ''f''<sub>''i''</sub> determines one coordinate of the [[image (mathematics)|range]] of ''f''. | ||
If ''V''′ is a variety contained in '''A'''<sup>''m''</sup>, we say that ''f'' is a ''regular map'' from ''V'' to ''V''′ if the range of ''f'' is contained in ''V''′. | If ''V''′ is a variety contained in '''A'''<sup>''m''</sup>, we say that ''f'' is a ''regular map'' from ''V'' to ''V''′ if the range of ''f'' is contained in ''V''′. | ||
The definition of the regular maps apply also to algebraic sets. | The definition of the regular maps apply also to algebraic sets. The regular maps are also called ''morphisms'', as they make the collection of all affine algebraic sets into a [[category theory|category]], where the objects are the affine algebraic sets and the [[morphism]]s are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. | ||
The regular maps are also called ''morphisms'', as they make the collection of all affine algebraic sets into a [[category theory|category]], where the objects are the affine algebraic sets and the [[morphism]]s are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. | |||
Given a regular map ''g'' from ''V'' to ''V''′ and a regular function ''f'' of ''k''[''V''′], then {{nowrap|''f'' ∘ ''g'' ∈ ''k''[''V'']}}. The map {{nowrap|''f'' → ''f'' ∘ ''g''}} is a [[ring homomorphism]] from ''k''[''V''′] to ''k''[''V'']. Conversely, every ring homomorphism from ''k''[''V''′] to ''k''[''V''] defines a regular map from ''V'' to ''V''′. This defines an [[equivalence of categories]] between the category of algebraic sets and the [[dual (category theory)|opposite category]] of the finitely generated [[reduced ring|reduced]] ''k''-algebras. This equivalence is one of the starting points of [[scheme theory]]. | Given a regular map ''g'' from ''V'' to ''V''′ and a regular function ''f'' of ''k''[''V''′], then {{nowrap|''f'' ∘ ''g'' ∈ ''k''[''V'']}}. The map {{nowrap|''f'' → ''f'' ∘ ''g''}} is a [[ring homomorphism]] from ''k''[''V''′] to ''k''[''V'']. Conversely, every ring homomorphism from ''k''[''V''′] to ''k''[''V''] defines a regular map from ''V'' to ''V''′. This defines an [[equivalence of categories]] between the category of algebraic sets and the [[dual (category theory)|opposite category]] of the finitely generated [[reduced ring|reduced]] ''k''-algebras. This equivalence is one of the starting points of [[scheme theory]]. | ||
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In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. | In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. | ||
If ''V'' is an affine variety, its coordinate ring is an [[integral domain]] and has thus a [[field of fractions]] which is denoted ''k''(''V'') and called the ''field of the rational functions'' on ''V'' or, shortly, the ''[[function field of an algebraic variety|function field]]'' of ''V''. Its elements are the restrictions to ''V'' of the [[rational function]]s over the affine space containing ''V''. The [[domain of a function|domain]] of a rational function ''f'' is not ''V'' but the [[complement (set theory)|complement]] of the | If ''V'' is an affine variety, its coordinate ring is an [[integral domain]] and has thus a [[field of fractions]] which is denoted ''k''(''V'') and called the ''field of the rational functions'' on ''V'' or, shortly, the ''[[function field of an algebraic variety|function field]]'' of ''V''. Its elements are the restrictions to ''V'' of the [[rational function]]s over the affine space containing ''V''. The [[domain of a function|domain]] of a rational function ''f'' is not ''V'' but the [[complement (set theory)|complement]] of the sub-variety (a hypersurface) where the denominator of ''f'' vanishes. | ||
As with regular maps, one may define a ''rational map'' from a variety ''V'' to a variety ''V''<nowiki>'</nowiki>. As with the regular maps, the rational maps from ''V'' to ''V''<nowiki>'</nowiki> may be identified to the [[ring homomorphism|field homomorphism]]s from ''k''(''V''<nowiki>'</nowiki>) to ''k''(''V''). | As with regular maps, one may define a ''rational map'' from a variety ''V'' to a variety ''V''<nowiki>'</nowiki>. As with the regular maps, the rational maps from ''V'' to ''V''<nowiki>'</nowiki> may be identified to the [[ring homomorphism|field homomorphism]]s from ''k''(''V''<nowiki>'</nowiki>) to ''k''(''V''). | ||
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which may also be viewed as a rational map from the line to the circle. | which may also be viewed as a rational map from the line to the circle. | ||
The problem of [[resolution of singularities]] is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is | The problem of [[resolution of singularities]] is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is non-singular (see also [[smooth completion]]). It was solved in the affirmative in [[Characteristic (algebra)|characteristic]] 0 by [[Heisuke Hironaka]] in 1964 and is yet unsolved in finite characteristic. | ||
=== Projective variety === | === Projective variety === | ||
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To see how this might come about, consider the variety {{nowrap|''V''(''y'' − ''x''<sup>2</sup>)}}. If we draw it, we get a [[parabola]]. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''<sup>2</sup>) also goes to positive infinity. As ''x'' goes to negative infinity, the slope of the same line goes to negative infinity. | To see how this might come about, consider the variety {{nowrap|''V''(''y'' − ''x''<sup>2</sup>)}}. If we draw it, we get a [[parabola]]. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''<sup>2</sup>) also goes to positive infinity. As ''x'' goes to negative infinity, the slope of the same line goes to negative infinity. | ||
Compare this to the variety ''V''(''y'' − ''x''<sup>3</sup>). This is a [[cubic curve]]. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''<sup>3</sup>) goes to positive infinity just as before. But unlike before, as ''x'' goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the | Compare this to the variety ''V''(''y'' − ''x''<sup>3</sup>). This is a [[cubic curve]]. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''<sup>3</sup>) goes to positive infinity just as before. But unlike before, as ''x'' goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behaviour "at infinity" of ''V''(''y'' − ''x''<sup>3</sup>) is different from the behaviour "at infinity" of ''V''(''y'' − ''x''<sup>2</sup>). | ||
The consideration of the ''projective completion'' of the two curves, which is their prolongation "at infinity" in the [[projective plane]], allows us to quantify this difference: the point at infinity of the parabola is a [[regular point of an algebraic variety|regular point]], whose tangent is the [[line at infinity]], while the point at infinity of the cubic curve is a [[cusp (singularity)|cusp]]. Also, both curves are rational, as they are parameterized by ''x'', and the [[Riemann-Roch theorem for algebraic curves|Riemann-Roch theorem]] implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular. | The consideration of the ''projective completion'' of the two curves, which is their prolongation "at infinity" in the [[projective plane]], allows us to quantify this difference: the point at infinity of the parabola is a [[regular point of an algebraic variety|regular point]], whose tangent is the [[line at infinity]], while the point at infinity of the cubic curve is a [[cusp (singularity)|cusp]]. Also, both curves are rational, as they are parameterized by ''x'', and the [[Riemann-Roch theorem for algebraic curves|Riemann-Roch theorem]] implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular. | ||
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The fact that the field of the real numbers is an [[ordered field]] cannot be ignored in such a study. For example, the curve of equation <math>x^2+y^2-a=0</math> is a circle if <math> a>0</math>, but has no real points if <math> a<0</math>. Real algebraic geometry also investigates, more broadly, ''[[semi-algebraic set]]s'', which are the solutions of systems of polynomial inequalities. For example, neither branch of the [[hyperbola]] of equation <math>x y-1 = 0</math> is a real algebraic variety. However, the branch in the first quadrant is a semi-algebraic set defined by <math>x y-1=0</math> and <math>x>0</math>. | The fact that the field of the real numbers is an [[ordered field]] cannot be ignored in such a study. For example, the curve of equation <math>x^2+y^2-a=0</math> is a circle if <math> a>0</math>, but has no real points if <math> a<0</math>. Real algebraic geometry also investigates, more broadly, ''[[semi-algebraic set]]s'', which are the solutions of systems of polynomial inequalities. For example, neither branch of the [[hyperbola]] of equation <math>x y-1 = 0</math> is a real algebraic variety. However, the branch in the first quadrant is a semi-algebraic set defined by <math>x y-1=0</math> and <math>x>0</math>. | ||
One open problem in real algebraic geometry is the following part of [[Hilbert's sixteenth problem]]: Decide which respective positions are possible for the ovals of a | One open problem in real algebraic geometry is the following part of [[Hilbert's sixteenth problem]]: Decide which respective positions are possible for the ovals of a non-singular [[plane curve]] of degree 8. | ||
== Computational algebraic geometry == | == Computational algebraic geometry == | ||
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==Applications== | ==Applications== | ||
Algebraic geometry now finds applications in [[algebraic statistics|statistics]],<ref>{{cite book| last1 = Drton| first1 = Mathias| last2 = Sturmfels| first2 = Bernd| last3 = Sullivant| first3 = Seth| title = Lectures on Algebraic Statistics| url = https://books.google.com/books?id=TytYUTy5V_IC| year = 2009| publisher = Springer| isbn = 978-3-7643-8904-8 }}</ref> [[control theory]],<ref>{{cite book| last = Falb| first = Peter| title = Methods of Algebraic Geometry in Control Theory Part II Multivariable Linear Systems and Projective Algebraic Geometry| url = https://books.google.com/books?id=V--84aGmWh4C| year = 1990| publisher = Springer| isbn = 978-0-8176-4113-9 }}</ref><ref>{{cite book |author-link=Allen Tannenbaum |first=Allen |last=Tannenbaum |date=1982 |title=Invariance and Systems Theory: Algebraic and Geometric Aspects |series=Lecture Notes in Mathematics |volume=845 |publisher=Springer-Verlag |isbn=9783540105657}}</ref> [[robotics]],<ref>{{cite book| last = Selig| first = J. M.| title = Geometric Fundamentals of Robotics| url = https://books.google.com/books?id=9FljXoISr8AC| year = 2005| publisher = Springer| isbn = 978-0-387-20874-9 }}</ref> [[algebraic geometric code|error-correcting codes]],<ref>{{cite book |last1=Tsfasman |first1=Michael A. |last2=Vlăduț |first2=Serge G. |last3=Nogin |first3=Dmitry |title=Algebraic Geometric Codes Basic Notions |year=1990 |publisher=American Mathematical Soc. |isbn=978-0-8218-7520-9 |url=https://books.google.com/books?id=o2sA-wzDBLkC}}</ref> [[computational phylogenetics|phylogenetics]]<ref>{{cite journal |author-link=Barry Arthur Cipra |first=Barry Arthur |last=Cipra |date=2007 |url=http://siam.org/pdf/news/1146.pdf |title=Algebraic Geometers See Ideal Approach to Biology |archive-url=https://web.archive.org/web/20160303230428/http://siam.org/pdf/news/1146.pdf |archive-date=3 March 2016 |journal=SIAM News |volume=40 |issue=6}}</ref> and [[geometric modelling]].<ref>{{cite book |last1=Jüttler |first1=Bert |last2=Piene |first2=Ragni |title=Geometric Modeling and Algebraic Geometry |year=2007 |publisher=Springer |isbn=978-3-540-72185-7 |url=https://books.google.com/books?id=1wNGq87gWykC}}</ref> There are also connections to [[Homological mirror symmetry|string theory]],<ref>{{cite book |last1=Cox |first1=David A. |author-link1=David A. Cox |last2=Katz |first2=Sheldon |title=Mirror Symmetry and Algebraic Geometry |url=https://books.google.com/books?id=Z8u3ngEACAAJ |year=1999 |publisher=American Mathematical Soc. |isbn=978-0-8218-2127-5}}</ref> [[game theory]],<ref>{{cite journal |title=The algebraic geometry of perfect and sequential equilibrium |first1=L. E. |last1=Blume |first2=W. R. |last2=Zame |journal=[[Econometrica]] |volume=62 |issue=4 |year=1994 |pages=783–794 |doi=10.2307/2951732 |jstor=2951732 }}</ref> [[Matching (graph theory)|graph matching]]s,<ref>{{cite arXiv |last1=Kenyon |first1=Richard |last2=Okounkov |first2=Andrei |last3=Sheffield |first3=Scott |title=Dimers and Amoebae |eprint=math-ph/0311005 |year=2003}}</ref> [[soliton]]s<ref>{{cite book |last=Fordy |first=Allan P. |title=Soliton Theory A Survey of Results |url=https://books.google.com/books?id=eO_PAAAAIAAJ |year=1990 |publisher=Manchester University Press |isbn=978-0-7190-1491-8}}</ref> and [[integer programming]].<ref>{{cite book |last1=Cox |first1=David A. |author-link1=David A. Cox |last2=Sturmfels |first2=Bernd |editor-last=Manocha |editor-first=Dinesh N. |title=Applications of Computational Algebraic Geometry |url=https://books.google.com/books?id=fe0MJEPDwzAC |publisher=American Mathematical Soc. |isbn=978-0-8218-6758-7}}</ref> | Algebraic geometry now finds applications in [[algebraic statistics|statistics]],<ref>{{cite book| last1 = Drton| first1 = Mathias| last2 = Sturmfels| first2 = Bernd| last3 = Sullivant| first3 = Seth| title = Lectures on Algebraic Statistics| url = https://books.google.com/books?id=TytYUTy5V_IC| year = 2009| publisher = Springer| isbn = 978-3-7643-8904-8 }}</ref> [[control theory]],<ref>{{cite book| last = Falb| first = Peter| title = Methods of Algebraic Geometry in Control Theory Part II Multivariable Linear Systems and Projective Algebraic Geometry| url = https://books.google.com/books?id=V--84aGmWh4C| year = 1990| publisher = Springer| isbn = 978-0-8176-4113-9 }}</ref><ref>{{cite book |author-link=Allen Tannenbaum |first=Allen |last=Tannenbaum |date=1982 |title=Invariance and Systems Theory: Algebraic and Geometric Aspects |series=Lecture Notes in Mathematics |volume=845 |publisher=Springer-Verlag |isbn=9783540105657}}</ref> [[robotics]],<ref>{{cite book| last = Selig| first = J. M.| title = Geometric Fundamentals of Robotics| url = https://books.google.com/books?id=9FljXoISr8AC| year = 2005| publisher = Springer| isbn = 978-0-387-20874-9 }}</ref> [[algebraic geometric code|error-correcting codes]],<ref>{{cite book |last1=Tsfasman |first1=Michael A. |last2=Vlăduț |first2=Serge G. |last3=Nogin |first3=Dmitry |title=Algebraic Geometric Codes Basic Notions |year=1990 |publisher=American Mathematical Soc. |isbn=978-0-8218-7520-9 |url=https://books.google.com/books?id=o2sA-wzDBLkC}}</ref> [[computational phylogenetics|phylogenetics]]<ref>{{cite journal |author-link=Barry Arthur Cipra |first=Barry Arthur |last=Cipra |date=2007 |url=http://siam.org/pdf/news/1146.pdf |title=Algebraic Geometers See Ideal Approach to Biology |archive-url=https://web.archive.org/web/20160303230428/http://siam.org/pdf/news/1146.pdf |archive-date=3 March 2016 |journal=SIAM News |volume=40 |issue=6}}</ref> and [[geometric modelling]].<ref>{{cite book |last1=Jüttler |first1=Bert |last2=Piene |first2=Ragni |title=Geometric Modeling and Algebraic Geometry |year=2007 |publisher=Springer |isbn=978-3-540-72185-7 |url=https://books.google.com/books?id=1wNGq87gWykC}}</ref> There are also connections to [[Homological mirror symmetry|string theory]],<ref>{{cite book |last1=Cox |first1=David A. |author-link1=David A. Cox |last2=Katz |first2=Sheldon |author-link2=Sheldon Katz |title=Mirror Symmetry and Algebraic Geometry |url=https://books.google.com/books?id=Z8u3ngEACAAJ |year=1999 |publisher=American Mathematical Soc. |isbn=978-0-8218-2127-5}}</ref> [[game theory]],<ref>{{cite journal |title=The algebraic geometry of perfect and sequential equilibrium |first1=L. E. |last1=Blume |first2=W. R. |last2=Zame |journal=[[Econometrica]] |volume=62 |issue=4 |year=1994 |pages=783–794 |doi=10.2307/2951732 |jstor=2951732 }}</ref> [[Matching (graph theory)|graph matching]]s,<ref>{{cite arXiv |last1=Kenyon |first1=Richard |last2=Okounkov |first2=Andrei |last3=Sheffield |first3=Scott |title=Dimers and Amoebae |eprint=math-ph/0311005 |year=2003 }}</ref> [[soliton]]s<ref>{{cite book |last=Fordy |first=Allan P. |title=Soliton Theory A Survey of Results |url=https://books.google.com/books?id=eO_PAAAAIAAJ |year=1990 |publisher=Manchester University Press |isbn=978-0-7190-1491-8}}</ref> and [[integer programming]].<ref>{{cite book |last1=Cox |first1=David A. |author-link1=David A. Cox |last2=Sturmfels |first2=Bernd |editor-last=Manocha |editor-first=Dinesh N. |title=Applications of Computational Algebraic Geometry |url=https://books.google.com/books?id=fe0MJEPDwzAC |publisher=American Mathematical Soc. |isbn=978-0-8218-6758-7}}</ref> | ||
==See also== | ==See also== | ||
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;Textbooks in computational algebraic geometry | ;Textbooks in computational algebraic geometry | ||
* {{cite book |last1=Cox |first1=David A. |author- | * {{cite book |last1=Cox |first1=David A. |author-link=David A. Cox |author-link2=John B. Little (mathematician) |author-link3=Donal O'Shea |last2=Little |first2=John |last3=O'Shea |first3=Donal |title=Ideals, Varieties, and Algorithms |edition=2nd |year=1997 |publisher=[[Springer Science+Business Media|Springer-Verlag]] |isbn=978-0-387-94680-1 |zbl=0861.13012}} | ||
*{{cite book | *{{cite book | ||
| last1=Schenck |first1=Hal |author-link1=Hal Schenck | | last1=Schenck |first1=Hal |author-link1=Hal Schenck | ||
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| url=https://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/computational-algebraic-geometry?format=HB&isbn=9780521829649 | publisher = [[Cambridge University Press]] | | url=https://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/computational-algebraic-geometry?format=HB&isbn=9780521829649 | publisher = [[Cambridge University Press]] | ||
}} | }} | ||
* {{cite book | * {{cite book | last1=Basu | first1=Saugata | last2=Pollack | first2=Richard | last3=Roy | first3=Marie-Françoise | year=2006 | title=Algorithms in real algebraic geometry | publisher=[[Springer Science+Business Media|Springer-Verlag]] | url=http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html | archive-date=2022-11-16 | access-date=2013-01-17 | archive-url=https://web.archive.org/web/20221116144409/https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html | url-status=dead }} | ||
}} | |||
* {{cite book | * {{cite book | ||
| last1=González-Vega |first1=Laureano | | last1=González-Vega |first1=Laureano | ||
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}} | }} | ||
* {{cite book | * {{cite book | ||
| last1=Cox |first1=David A. |author- | | last1=Cox |first1=David A. |author-link=David A. Cox |author-link2=John B. Little (mathematician) |author-link3=Donal O'Shea | ||
| last2=Little |first2=John B. | | last2=Little |first2=John B. | ||
| last3=O'Shea |first3=Donal | | last3=O'Shea |first3=Donal | ||