* Tate, John ( 1963 ), " Algebraic cycles and poles of zeta functions ", in Arithmetical Algebraic Geometry, Harper and Row: 93-110, MR0225778.

Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions.

where the coefficients are polynomial functions of x. Algebraic functions play an important role in mathematical analysis and algebraic geometry.

Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation z = w < sup > 2 </ sup > for w as a function of z.

Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces.

* Algebraic functions, functions satisfying certain polynomials

Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions ( non-algebraic functions ) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.

* Algebraic code-excited linear prediction ( ACELP 4. 7 kbit / s – 24 kbit / s )

; Algebraic extension: If an element α of an extension field E over F is the root of a non-zero polynomial in F, then α is algebraic over F. If every element of E is algebraic over F, then E / F is an algebraic extension.

* Vakil, Ravi, " Foundations of Algebraic Geometry ", Lecture Notes, http :// math. stanford. edu /~ vakil / 216blog /

His areas of expertise were: Category theory, Abelian categories with Applications to Rings and Modules, adjoint functors, limits / colimits ,, Theory of Sheaves, Theory of Rings, Fields and Polynomials, and Valuation Theory ; he also had interests and published in the following areas: Algebraic Topology, Algebraic Geometry, Commutative Algebra, K-Theory, Class-Field theory, and Algebraic Function Theory.

* Algebraic Logic Functional programming language, a functional / logic programming language.

Algebraic K-groups are used in conjectures on special values of L-functions and the formulation of an non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.

Non-Abelian Algebraic Topology: filtered spaces, crossed complexes, cubical higher homotopy groupoids ; European Mathematical Society Tracts in Mathematics Vol.

* Algebraic simplification ; for example, the CAS can combine multiple terms into one fraction by finding a common denominator.

Algebraic notation is more concise and requires less effort to avoid ambiguity ; however much older literature uses descriptive notation.

This includes move number indicators ( numbers followed by either one or three periods ; one if the next move is White's move, three if the next move is Black's move ) and movetext Standard Algebraic Notation ( SAN ).

The algebraic name of any square is as per usual Algebraic chess notation ; from white's perspective, the leftmost square closest to white is a1, the rightmost square closest to the white is h1, and the rightmost ( from white's perspective ) square closest to black side is h8.

; Algebraic closure: An algebraic closure of a field F is an algebraic extension of F which is algebraically closed.

; Algebraic numbers: The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers.

; Algebraic reasoning

* Brown, R., Higgins, P. J. and Sivera, R .. 2011, EMS Tracts in Mathematics Vol. 15 ( 2011 ) Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids ; ( The first of three Parts discusses the applications of the 1-and 2-dimensional versions of the Seifert-van Kampen Theorem.

* Algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner ;

* Victor Batyrev ; Dual Polyhedra and for Calabi-Yau Hypersurfaces in Toric Varieties J. Algebraic Geom.

; Algebraic:

" It contained ideas of his teacher, Artin ; some of the most interesting passages in Algebraic Number Theory also reflect Artin's influence and ideas that might otherwise not have been published in that or any form.

The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space ; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points.

In order to get a statement in more classical language, but still wider than Serre duality, Hartshorne ( Algebraic Geometry ) uses the Ext functor of sheaves ; this is a kind of stepping stone to the derived category.

* Groupes Algébriques et Corps de Classes ( 1959 ), translated in English as Algebraic Groups and Class Fields ( 1988 )

Algebraic numbers coloured by degree.

Algebraic numbers coloured by leading coefficient ( red signifies 1 for an algebraic integer ).

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article, " Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen " (" On a Property of the Collection of All Real Algebraic Numbers ").

Algebraic rules were given geometric proofs by mathematicians such as Pacioli, Cardan, Tartaglia and Ferrari.

Algebraic notation is based on a system developed by Philipp Stamma.

This led the foundation to another powerful reconstruction method called " Algebraic Reconstruction Technique ( ART )" which was later adapted by Sir Godfrey Hounsfield as the image reconstruction mechanism in his famous invention, the first commercial CT scanner.

Developed in 1959 at the University of Michigan by Bernard Galler, Bruce Arden and Robert M. Graham, MAD is a variant of the International Algebraic Language ( IAL ).

Following the Standard Algebraic Notation ( SAN ), each piece is identified by a single letter taken from the standard English names ( pawn

The SART algorithm ( Simultaneous Algebraic Reconstruction Technique ) proposed by Andersen and Kak in 1984 has had a major impact in CT imaging

The concept of an exact sequence only appeared in print in the 1952 book Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod where the results of Mayer and Vietoris were expressed in the modern form.

* Algebraic Curves: An Introduction To Algebraic Geometry, by William Fulton with Richard Weiss.

The Éléments de géométrie algébrique (" Elements of Algebraic Geometry ") by Alexander Grothendieck ( assisted by Jean Dieudonné ), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published ( in eight parts or fascicles ) from 1960 through 1967 by the Institut des Hautes Études Scientifiques.

Algebraic tori form an important class of commutative group schemes, defined either by the property of being locally on S a product of copies of G < sub > m </ sub >, or as groups of multiplicative type associated to finitely generated free abelian groups.

* John Stillwell, introduction to Theory of Algebraic Integers by Richard Dedekind.

* Use Figurine Algebraic Notation, which replaces the letter that stands for a piece by its symbol, e. g. instead of Nc6.

Algebraic number theory is both the study of number theory by algebraic methods and the theory of algebraic numbers.

Algebraic code-excited linear prediction ( ACELP ) is a patented speech coding algorithm by VoiceAge Corporation in which a limited set of pulses is distributed as excitation to linear prediction filter.

* Algebraic Approaches to Program Semantics ( Texts and Monographs in Computer Science ) by Ernest G. Manes, Michael A. Arbib ( August 1, 1986 )

* Algebraic Systems by A. I.