The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.Communicated by James Renegar.

Abstract. This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective p-secant varieties to Segre varieties for p ≤ 6. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of Pn. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P1 × P1 × Pn × Pn and P2 × P3 × P3. In the final section we propose a series of conjectures about defective Segre varieties.

The Alexander-Hirschowitz theorem says that a general collection of k double points in P n imposes independent conditions on homogeneous polynomials of degree d with a well-known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on the previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d = 3, where our proof is shorter. We end with an account of the history of the work on this problem.

Abstract. We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank 1. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type (a, b, c), a ≤ b ≤ c, our method proves that the decomposition is unique (i.e. k-identifiability holds) for general tensors of rank k, as soon as k ≤ (a + 1)(b + 1)/16. This improves considerably the known range for identifiability. The method applies also to tensor of higher dimension. For tensors of small size, we give a complete list of situations where identifiability does not hold. Among them, there are 4 × 4 × 4 tensors of rank 6, an interesting case because of its connection with the study of DNA strings.

We propose a new sufficient condition for verifying whether general rank-r complex tensors of arbitrary order admit a unique decomposition as a linear combination of rank-1 tensors. A practical algorithm is proposed for verifying this condition, with which it was established that in all spaces of dimension less than 15000, with a few known exceptions, listed in the paper, generic identifiability holds for ranks up to one less than the generic rank of the space. This is the largest possible rank value for which generic identifiability can hold, except for spaces with a perfect shape. The algorithm can also verify the identifiability of a given specific rank-r decomposition, provided that it can be shown to correspond to a nonsingular point of the rth order secant variety. For sufficiently small rank, which nevertheless improves upon the known bounds for specific identifiability, some local equations of this variety are known, allowing us to verify this property. As a particular example of our approach, we prove the identifiability of a specific 5 × 5 × 5 tensor of rank 7, which cannot be handled by the conditions recently provided in [I. Domanov and L. De Lathauwer, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 876-903]. Finally, we also present a surprising new class of weakly defective Segre varieties that nevertheless turns out to admit a generically unique decomposition.