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We provide the first polynomial time exact algorithm for computing an Arrow-Debreu market equilibrium for the case of linear utilities. Our algorithm is based on solving a convex program using the ellipsoid algorithm and simultaneous diophantine approximation. As a side result, we prove that the set of assignments at equilibrium is convex and the equilibrium prices themselves are log-convex. Our convex program is explicit and intuitive, which allows maximizing a concave function over the set of equilibria. On the practical side, Ye developed an interior point algorithm [Lecture Notes in Comput. Sci. 3521, Springer, New York, 2005, pp. 3-5] to find an equilibrium based on our convex program. We also derive separate combinatorial characterizations of equilibrium for Arrow-Debreu and Fisher cases. Our convex program can be extended for many nonlinear utilities and production models. Our paper also makes a powerful theorem (Theorem 6.4.1 in [M. Grotschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 1993]) even more powerful (in Theorems 12 and 13) in the area of geometric algorithms and combinatorial optimization. The main idea in this generalization is to allow ellipsoids to contain not the whole convex region but a part of it. This theorem is of independent interest.