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We investigate the computational complexity of tensor rank, a concept thatplays fundamental role in different topics of modern applied mathematics. Fortensors over any integral domain, we prove that the rank problem is polynomialtime equivalent to solving a system of polynomial equations over this integraldomain. Our result gives a complete description of the algorithmic complexityof tensor rank and allows one to solve several known open problems. Inparticular, the tensor rank over $\mathbb{Z}$ turns out to be undecidable,which answers the question posed by Gonzalez and Ja'Ja' in 1980. We generalizeour result and prove that the symmetric rank admits a similar description ofcomputational complexity as the one we give for usual rank. In particular,computing the symmetric rank of a rational tensor is shown to be NP-hard, whichproves a recent conjecture of Hillar and Lim. As a byproduct of our approach,we get a similar characterization of the algorithmic complexity of the minimalrank matrix completion problem, which gives a complete answer to the questiondiscussed in 1999 by Buss, Frandsen, and Shallit.

How hard is the tensor rank?