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Determinants of Latin squares of order 8. (English) Zbl 0876.05017
Summary: A Latin square is an \(n\times n\) array of \(n\) symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-\(n\) polynomial in \(n\) variables. Can two Latin squares \(L\), \(M\) have the same determinant, up to a renaming of the variables, apart from the obvious cases when \(L\) is obtained from \(M\) by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to be no if \(n\leq 7\); we show that it is yes for \(n=8\). The Latin squares for which this situation occurs have interesting special characteristics.

05B15 Orthogonal arrays, Latin squares, Room squares
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