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Expressing a general form as a sum of determinants. (English) Zbl 1317.15006

Take a form \(F\in \mathbb{C}[x_1,\dots,x_n]\) and a \(k\times k\) integer matrix \(A\) such that \(\mathrm{deg}F=\mathrm{tr}A\). The question is whether \(F\) can be represented as a sum of determinants of matrices of type \(M=(F_{i,j})\), where \(\mathrm{deg} F_{i,j}=a_{i,j}\). In the cases \(n=2,3\), the answer is positive, and for \(n>4\) the answer is negative.
The matrix \(A\) is called homogeneous if \(a_{i,j}+a_{k,l}=a_{i,l}+a_{k,j}\). The main result of the paper is a theorem that states that if the matrix \(A\) is homogeneous, then the general form is a sum of \(s\) determinants of matrices of forms \((F_{l,m}^i)\) such that \(\mathrm{deg}(F_{l,m}^i)=a_{l,m}\). Estimates for \(s\) are given.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A03 Vector spaces, linear dependence, rank, lineability

Software:

SINGULAR
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References:

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