Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals
Résumé
Polynomial matrices and ideals generated by their minors appear in
various domains such as cryptography, polynomial optimization and
effective algebraic geometry. When the given matrix is symmetric, this
additional structure on top of the determinantal structure, affects
computations on the derived ideals. Thus, understanding the
complexity of these computations is important. Moreover, this study
serves as a stepping stone towards further understanding the effects
of structure in determinantal systems, such as those coming from
moment matrices. In this paper, we focus on the Sparse-FGLM
algorithm, the state-of-the-art for changing ordering of Gr\"obner
bases of zero-dimensional ideals. Under a variant of Fr\"oberg's
conjecture, we study its complexity for symmetric determinantal ideals
and identify the gain of exploiting sparsity in the Sparse-FGLM
algorithm compared with the classical FGLM algorithm. For an $n\times
n$ symmetric matrix with polynomial entries of degree $d$, we show
that the complexity of Sparse-FGLM for zero-dimensional determinantal
ideals obtained from this matrix over that of the FGLM algorithm is at
least $O(1/d)$. Moreover, for some specific sizes of minors, we prove
finer results of at least $O(1/nd)$ and $O(1/n^3d)$.
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