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91 relations: Abelian group, Abelian variety, Affine variety, Algebraic curve, Algebraic geometry, Algebraic group, Algebraic number field, Algebraic variety, Algebraically closed field, Algorithm, American Mathematical Society, André Weil, Andrew Wiles, Annales de l'Institut Fourier, Arithmetic dynamics, Associative algebra, Beniamino Segre, Birational geometry, Birch and Swinnerton-Dyer conjecture, Birch's theorem, Cambridge University Press, Chevalley–Warning theorem, Christopher Hooley, Commutative ring, Complex number, Conic section, Crelle's Journal, Cubic surface, Diophantine equation, Diophantine geometry, Element (category theory), Elliptic curve, Elliptic surface, Enrico Bombieri, Ernst Sejersted Selmer, Faltings's theorem, Fano variety, Fermat's Last Theorem, Fiber product of schemes, Field (mathematics), Field extension, Finite field, Finitely generated abelian group, Finnish Academy of Science and Letters, Functor, Genus (mathematics), Glossary of arithmetic and diophantine geometry, Group of rational points on the unit circle, Hardy–Littlewood circle method, Hasse principle, ..., Hasse–Minkowski theorem, Hélène Esnault, Homogeneous polynomial, Hypersurface, Integer, Inventiones Mathematicae, J. W. S. Cassels, János Kollár, Jean-Louis Colliot-Thélène, K3 surface, Kodaira dimension, Manin conjecture, Manin obstruction, Mordell–Weil theorem, Morphism of schemes, Number theory, Orbifold, P-adic number, Perfect field, Pierre Deligne, Polynomial, Projective cone, Projective space, Projective variety, Pythagorean triple, Rational number, Rational variety, Real number, Richard Taylor (mathematician), Roger Heath-Brown, Scheme (mathematics), Section (category theory), Serge Lang, Smooth scheme, Spectrum of a ring, Springer Nature, Tate–Shafarevich group, Unit circle, Yuri Manin, Zariski topology, Zero of a function. Expand index (41 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.
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Affine variety
In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.
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Algebraic curve
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic group
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry.
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Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
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Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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André Weil
André Weil (6 May 1906 – 6 August 1998) was an influential French mathematician of the 20th century, known for his foundational work in number theory, algebraic geometry.
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Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory.
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Annales de l'Institut Fourier
The Annales de l'Institut Fourier is a French mathematical journal publishing papers in all fields of mathematics.
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Arithmetic dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory.
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Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
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Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
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Birational geometry
In mathematics, birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets.
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Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve.
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Birch's theorem
In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Chevalley–Warning theorem
In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions.
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Christopher Hooley
Christopher Hooley FLSW FRS (born 7 August 1928) is a British mathematician, emeritus professor of mathematics at Cardiff University.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Conic section
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
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Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).
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Cubic surface
A cubic surface is a projective variety studied in algebraic geometry.
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Diophantine equation
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).
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Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers.
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Element (category theory)
In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category.
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Elliptic curve
In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.
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Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1.
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Enrico Bombieri
Enrico Bombieri (born 26 November 1940 in Milan) is an Italian mathematician, known for his work in analytic number theory, algebraic geometry, univalent functions, theory of several complex variables, partial differential equations of minimal surfaces, and the theory of finite groups.
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Ernst Sejersted Selmer
Ernst Sejersted Selmer (20 February 1920 – 8 November 2006) was a Norwegian mathematician who worked on number theory.
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Faltings's theorem
In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.
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Fano variety
In algebraic geometry, a Fano variety, introduced in, is a complete variety X whose anticanonical bundle KX* is ample.
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Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.
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Fiber product of schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field extension
In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
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Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
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Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.
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Finnish Academy of Science and Letters
The Finnish Academy of Science and Letters (Finnish Suomalainen Tiedeakatemia; Latin Academia Scientiarum Fennica) is a Finnish learned society.
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Functor
In mathematics, a functor is a map between categories.
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Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings.
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Glossary of arithmetic and diophantine geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.
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Group of rational points on the unit circle
In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2.
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Hardy–Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory.
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Hasse principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number.
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Hasse–Minkowski theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field (which may be real, complex, or p-adic).
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Hélène Esnault
Hélène Esnault (born 1953 in Paris) is a French and German mathematician, specializing in algebraic geometry.
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Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.
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Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Inventiones Mathematicae
Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.
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J. W. S. Cassels
John William Scott "Ian" Cassels, FRS (11 July 1922 – 27 July 2015) was a British mathematician.
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János Kollár
János Kollár (born June 7, 1956) is a Hungarian mathematician, specializing in algebraic geometry.
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Jean-Louis Colliot-Thélène
Jean-Louis Colliot-Thélène (born 2 December 1947), is a French mathematician.
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K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.
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Kodaira dimension
In algebraic geometry, the Kodaira dimension κ(X) (or canonical dimension) measures the size of the canonical model of a projective variety X. Igor Shafarevich introduced an important numerical invariant of surfaces with the notation κ in the seminar Shafarevich 1965.
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Manin conjecture
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function.
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Manin obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety X over a global field, which measures the failure of the Hasse principle for X. If the value of the obstruction is non-trivial, then X may have points over all local fields but not over the global field.
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Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group.
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Morphism of schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.
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P-adic number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.
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Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.
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Pierre Deligne
Pierre René, Viscount Deligne (born 3 October 1944) is a Belgian mathematician.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Projective cone
A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R. In the special case that R is a single point, S is a plane, and A is a conic section on S, the projective cone is a conical surface; hence the name.
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Projective space
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
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Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.
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Pythagorean triple
A Pythagorean triple consists of three positive integers,, and, such that.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set \ of indeterminates, where d is the dimension of the variety.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Richard Taylor (mathematician)
Richard Lawrence Taylor (born 19 May 1962) is a British and American mathematician working in the field of number theory.
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Roger Heath-Brown
David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory.
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
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Section (category theory)
In category theory, a branch of mathematics, a section is a right inverse of some morphism.
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Serge Lang
Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.
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Smooth scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point.
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Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
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Springer Nature
Springer Nature is an academic publishing company created by the May 2015 merger of Springer Science+Business Media and Holtzbrinck Publishing Group's Nature Publishing Group, Palgrave Macmillan, and Macmillan Education.
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Tate–Shafarevich group
In arithmetic geometry, the Tate–Shafarevich group Ш(A/K), introduced by and, of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group WC(A/K).
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Unit circle
In mathematics, a unit circle is a circle with a radius of one.
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Yuri Manin
Yuri Ivanovitch Manin (Ю́рий Ива́нович Ма́нин; born 1937) is a Soviet/Russian/German CURRICULUM VITAE at Max-Planck-Institut für Mathematik website mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.
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Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.
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Zero of a function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
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Redirects here:
K-rational point, Rational points.
References
[1] https://en.wikipedia.org/wiki/Rational_point
