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for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals. For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide, but for other fields there may exist
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algebraic matroids that are not linear; indeed the non-Pappus matroid is algebraic over any finite field, but not linear and not algebraic over any field of characteristic zero. However, if a matroid is algebraic over a field
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141:, in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is
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117:. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the
658:
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97:. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set
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The class of algebraic matroids is closed under truncation and matroid union. It is not known whether the
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of an algebraic matroid is always algebraic and there is no excluded minor characterisation of the class.
650:
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can be used to show that there always exists a maximal algebraically independent subset of
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Lindström, Bernt (1985). "On the algebraic characteristic set for a class of matroids".
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649:, Encyclopedia of Mathematics and its Applications, vol. 29, Cambridge:
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70:. Further, all the maximal algebraically independent subsets have the same
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Combinatorial
Mathematics and its Applications (Proc. Conf., Oxford, 1969)
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218:
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145:. Every matroid with a linear representation of this type over a field
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Ingleton, A. W.; Main, R. A. (1975). "Non-algebraic matroids exist".
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209:. It follows that the class of algebraic matroids is closed under
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Every prime occurs as the unique characteristic for some matroid.
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structure, that expresses an abstraction of the relation of
109:. A matroid that can be generated in this way is called
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satisfy the axioms that define the independent sets of a
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Ingleton, A. W. (1971). "Representation of matroids".
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may also be represented as an algebraic matroid over
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540:Proceedings of the American Mathematical Society
162:of characteristic zero then it is linear over
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444:. London: Academic Press. pp. 149–167.
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275:) then all sufficiently large primes are in
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170:) for some finite set of transcendentals
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194:If a matroid is algebraic over a
263:is algebraically representable.
626:. Courier Dover Publications.
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242:(algebraic) characteristic set
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305:and hence so is any minor of
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205:) then it is algebraic over
129:Many finite matroids may be
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473:Applied Discrete Structures
125:Relation to linear matroids
115:algebraically representable
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651:Cambridge University Press
641:White, Neil, ed. (1987),
645:Combinatorial geometries
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297:then any contraction of
594:Oxford University Press
255:is the set of possible
217:is algebraic over the
37:algebraic independence
471:Joshi, K. D. (1997),
259:of fields over which
81:For every finite set
369:10.1112/blms/7.2.144
143:linearly independent
76:transcendence degree
527:Oxley (1992) p.223
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410:Oxley (1992) p.220
401:Oxley (1992) p.221
343:Oxley (1992) p.215
334:Oxley (1992) p.218
325:Oxley (1992) p.216
301:is algebraic over
293:is algebraic over
236:Characteristic set
190:Closure properties
78:of the extension.
518:White (1987) p.25
430:White (1987) p.24
180:algebraic closure
153:, by choosing an
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267:If 0 is in
219:prime field
211:contraction
131:represented
72:cardinality
21:mathematics
669:0626.00007
612:0784.05002
579:References
569:0572.05019
458:0222.05025
385:0315.05018
43:Definition
622:(2010) .
111:algebraic
679:Category
588:(1992).
47:Given a
561:2045591
450:0278974
377:0369110
95:matroid
29:matroid
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135:matrix
557:JSTOR
314:Notes
174:over
133:by a
66:over
27:is a
23:, an
655:ISBN
628:ISBN
598:ISBN
477:ISBN
240:The
230:dual
31:, a
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