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Summary: AN ELIMINATION THEOREM FOR MIXED REAL-INTEGER
SYSTEMS
MATTHIAS ASCHENBRENNER
Abstract. An elimination result for mixed real-integer systems of linear equa-
tions is established, and used to give a short proof for an adaptation of Farkas'
Lemma by KĻoppe and Weismantel [4]. An extension of the elimination theo-
rem to a quantifier elimination result is indicated.
Let A be an m Ũ n-matrix with real entries and b Rm
. (In the following, we
think of elements of the various euclidean spaces Rk
as column vectors.) If b is not
contained in the closed convex cone C generated by the columns of A, then b can
be separated from C by a hyperplane. More precisely, the following are equivalent:
(1) There is no column vector x = (x1, . . . , xn)t
Rn
with Ax = b and x 0
(i.e., xi 0 for every i = 1, . . . , n).
(2) There is some y Rm
such that yt
A 0 and yt
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