163:, may be considered to belong to elimination theory, as it asserts that a system of polynomial equations does not have any solution if and only if one may eliminate all unknowns to obtain the constant equation 1 = 0.
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introduced non-effective methods, and this was seen as a revolution, which led most algebraic geometers of the first half of the 20th century to try to "eliminate elimination". Nevertheless
210:, which again made relevant the design of efficient elimination algorithms, rather than merely existence and structural results. The main methods for this renewal of elimination theory are
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does not proceed by elimination, and works only when the number of equations equals the number of variables. In the 19th century, this was extended to linear
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to explain that, in some theories, every formula is equivalent to a formula without quantifier. This is the case of the theory of
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All these concepts are effective, in the sense that their definitions include a method of computation. Around 1890,
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Later, elimination theory was considered old-fashioned and removed from subsequent editions of
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is the classical name for algorithmic approaches to eliminating some variables between
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Part of algebraic geometry devoted to the elimination of variables between polynomials
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The field of elimination theory was motivated by the need of methods for solving
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variables for reducing the problem to a single equation in one variable.
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There is also a logical facet to elimination theory, as seen in the
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under various changes of variables, and are also fundamental in
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Discriminants, resultants, and multidimensional determinants
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Except for BĂ©zout's theorem, the general approach was to
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Classical elimination theory culminated with the work of
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The case of linear equations was completely solved by
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166:Elimination theory culminated with the work of
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250:Quantifier elimination over the reals
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216:cylindrical algebraic decomposition
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286:Main theorem of elimination theory
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271:Faugère's F4 and F5 algorithms
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161:Hilbert's Nullstellensatz
353:Using Algebraic Geometry
281:Triangular decomposition
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176:multivariate resultants
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266:Buchberger's algorithm
233:Quantifier elimination
56:Bartel van der Waerden
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111:Diophantine equations
300:, Mikhail Kapranov,
103:Gaussian elimination
222:Connection to logic
119:Hermite normal form
21:commutative algebra
384:Algebraic geometry
238:mathematical logic
236:is a term used in
184:Elimination theory
52:Elimination theory
29:elimination theory
25:algebraic geometry
369:978-0-387-20733-9
337:978-0-387-95385-4
302:Andrey Zelevinsky
188:van der Waerden's
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168:Leopold Kronecker
123:Smith normal form
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389:Computer algebra
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208:computer algebra
150:invariant theory
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320:Lang, Serge
242:polynomials
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378:Categories
292:References
135:resultants
130:eliminants
276:Resultant
204:computers
146:invariant
96:eliminate
322:(2002),
260:See also
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172:Macaulay
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324:Algebra
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