Abstract
We prove the existence of many homogeneous polynomials with non-unique minimal additive decompositions. For all integers n, d, k such that \(n\ge 2\), \(d\ge 6\) and \(\lceil d/2\rceil < k\le \left( {\begin{array}{c}n+\lfloor d/2\rfloor \\ n\end{array}}\right) \) there is a degree d form in \(n+1\) variables with rank k and having infinitely many decompositions as the sum of kd-powers of linear forms. We also construct different families of forms f and for each of them we compute the dimension (or the number) of its additive decompositions.
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E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).
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Ballico, E. Large families of homogeneous polynomials with non-unique additive decompositions. Beitr Algebra Geom 61, 35–45 (2020). https://doi.org/10.1007/s13366-019-00453-y
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DOI: https://doi.org/10.1007/s13366-019-00453-y