Geodesic
New polynomial identities for determinants over commutative rings
The Bulletin of Irkutsk State University. Series Mathematics, Tome 5 (2012) no. 4, pp. 16-20

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Let $K$ be a commutative ring with division by integers. Here we give a new family of polynomial identities (calculation formulas) for determinants over the ring $K$ using the well-known polarization theorem, which allows us a new criterian for linear independence of $n$ vectors in $\mathbb{C}^{n}$.
Keywords: determinants; commutative rings; polynomial identities. 
@article{IIGUM_2012_5_4_a1,
     author = {G. P. Egorychev},
     title = {New polynomial identities for determinants over commutative rings},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {16--20},
     publisher = {mathdoc},
     volume = {5},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2012_5_4_a1/}
}
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G. P. Egorychev. New polynomial identities for determinants over commutative rings. The Bulletin of Irkutsk State University. Series Mathematics, Tome 5 (2012) no. 4, pp. 16-20. http://geodesic.mathdoc.fr/item/IIGUM_2012_5_4_a1/