On the Alexander–Hirschowitz theorem

By Maria Chiara Brambilla, Giorgio Ottaviani. 2007

165 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ebfad908c9e63582ded4d2bca3bbee572de992dc

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In Computational Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

ON THE LENGTH OF BINARY FORMS (2010, background): “In 1995, Alexander and Hirschowitz [1] (see also [2, 31]) established that for n, d ≥ 3, this length is ⌈ 1 n ( n+d−1 n−1 ) ⌉, the Date: August 2, 2010.”
Stratification of the fourth secant variety of Veronese varieties via the symmetric rank (2010, background): “Hirschowitz in [2] (see also [7] for a modern proof).”
Moment Varieties of Gaussian Mixtures (2015, background): “[4]), where these secant varieties do not have the expected dimension.”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “The following historical review can also be found in [37].”
Classical Algebraic Geometry: A Modern View (2012, background): “A simplified proof can be found in [53] or [91].”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “4 of [7] (in P) on the trace (right space); it turns out to be empty.”
Non-defectivity of Grassmannians of planes (2009, methodology): “This technique was inspired by [ BO ], where the case X = (Pn;O(3)) was treated with a”
On the typical rank of real binary forms (2009, background): “The Theorem of Alexander and Hirschowitz [2] (see also [5]) answers that the generic rank is always the expected one, with a finite list of exceptions.”
An Invariant Regarding Waring’s Problem for Cubic Polynomials (2007, background): “Our starting point is the theorem of Alexander and Hirschowitz (see [AH] or [ BO ] for a survey, including a self-contained proof) which states that the codimension of �s(Pn, O(d)) ⊆ PN is the expected one, that is max{N + 1 − (n + 1)s,0}, with the”
Tensor decompositions, alternating least squares and other tales (2009, background): “Withthehelpoftheso-calledTerracini’slemma[43],sometimes attributed to Lasker [38,44], it is possible to compute the smallest typical rank for any given order and dimensions, since it coincides with the generic rank (if the decomposition were allowed to be computed in the complex ﬁeld) [35,45,46].”