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Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in theoretical computer science and machine learning. DPPs define probability distributions over subsets of a given ground set, they exhibit interesting properties such as negative correlation, and, unlike other models of negative correlation such as Markov random fields, have efficient algorithms for sampling. When applied to kernel methods in machine learning, DPPs favor subsets of the given data with more diverse features. However, many real-world applications require efficient algorithms to sample from DPPs with additional constraints on the sampled subset, e.g., partition or matroid constraints that are important from the viewpoint of ensuring priors, resource or fairness constraints on the sampled subset. Whether one can efficiently sample from DPPs in such constrained settings is an important problem that was first raised in a survey of DPPs for machine learning by Kulesza and Taskar and studied in some recent works. The main contribution of this paper is the first resolution of the complexity of sampling from DPPs with constraints. On the one hand, we give exact efficient algorithms for sampling from constrained DPPs when the description of the constraints is in unary; this includes special cases of practical importance such as a small number of partition, knapsack or budget constraints. On the other hand, we prove that when the constraints are specified in binary, this problem is #P-hard via a reduction from the problem of computing mixed discriminants; implying that it may be unlikely that there is an FPRAS. Technically, our algorithmic result benefits from viewing the constrained sampling problem via the lens of polynomials and we obtain our complexity results by providing an equivalence between computing mixed discriminants and sampling from partition constrained DPPs. As a consequence, we obtain a few corollaries of independent interest: 1) An algorithm to count, sample (and, hence, optimize) over the base polytope of regular matroids when there are additional (succinct) budget constraints and, 2) An algorithm to evaluate and compute mixed characteristic polynomials, that played a central role in the resolution of the Kadison-Singer problem, for certain special cases.