TY - JOUR

T1 - Degrees of convex dependence in recursively enumerable vector spaces

AU - Nevins, Thomas A.

N1 - Funding Information:
Correspondence to: T.A. Nevins, University of Notre Dame, 302 St. Edward’s Hall, Notre Dame, IN 46556, USA. Email: tnevins@elendil.next.nd.edu. * This work was supported by National Science Foundation grant DMS 92-00388. The author would like to thank Michael Stob for his encouragement, advice, and help with the mathematics contained in, and the writing of, this paper.

PY - 1993/2/24

Y1 - 1993/2/24

N2 - Let W be a recursively enumerable vector space over a recursive ordered field. We show the Turing equivalence of the following sets: the set of all tuples of vectors in W which are linearly dependent; the set of all tuples of vectors in W whose convex closures contain the zero vector; and the set of all pairs (X, Y) of tuples in W such that the convex closure of X intersects the convex closure of Y. We also form the analogous sets consisting of tuples with given numbers of elements, and prove similar results on the Turing equivalence of these.

AB - Let W be a recursively enumerable vector space over a recursive ordered field. We show the Turing equivalence of the following sets: the set of all tuples of vectors in W which are linearly dependent; the set of all tuples of vectors in W whose convex closures contain the zero vector; and the set of all pairs (X, Y) of tuples in W such that the convex closure of X intersects the convex closure of Y. We also form the analogous sets consisting of tuples with given numbers of elements, and prove similar results on the Turing equivalence of these.

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U2 - 10.1016/0168-0072(93)90191-F

DO - 10.1016/0168-0072(93)90191-F

M3 - Article

AN - SCOPUS:43949173111

SN - 0168-0072

VL - 60

SP - 31

EP - 47

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 1

ER -