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A short review of Algebraic Geometry tools for the decomposition of tensorsand polynomials is given from the point of view of applications to quantum andatomic physics. Examples of application to assemblies of indistinguishabletwo-level bosonic atoms are discussed using modern formulations of theclassical Sylvester's algorithm for the decomposition of homogeneouspolynomials in two variables. In particular, the symmetric rank and symmetricborder rank of spin squeezed states is calculated as well as theirSchr\"odinger-cat-like decomposition as the sum of macroscopically differentcoherent spin states; Fock states provide an example of states for which thesymmetric rank and the symmetric border rank are different.

Algebraic Geometry tools for the study of entanglement: an application to spin squeezed states