Computing loci of rank defects of linear matrices (also called the MinRank problem) is a fundamental NP-hard problem of linear algebra which has applications in Cryptology, in Error Correcting Codes and in Geometry. Given a square linear matrix (i.e. a matrix whose entries are k-variate linear forms) of size n and an integer r, the problem is to find points such that the evaluation of the matrix has rank less than r + 1. The aim of the paper is to obtain the most efficient algorithm to solve this problem. To this end, we give the theoretical and practical complexity of computing Gröbner bases of two algebraic formulations of the MinRank problem. Both modelings lead to structured algebraic systems. The first modeling, proposed by Kipnis and Shamir generates bihomogeneous equations of bi-degree (1,1). The second one is classically obtained by the vanishing of the (r + 1)-minors of the give
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