Geodesic
On the Gibson Bounds over Finite Fields
Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 395-416.

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We investigate the Pólya problem on the sign conversion between the permanent and the determinant over finite fields. The main attention is given to the sufficient conditions which guarantee non-existence of sing-conversion. In addition we show that F3 is the only field with the property that any matrix with the entries from the field is convertible. As a result we obtain that over finite fields there are no analogs of the upper Gibson barrier for the conversion and establish the lower convertibility barrier.
Keywords: Permanent, Determinant, Finite Fields, Pólya Problem 
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V. Budrevich, Mikhail; E. Guterman, Alexander. On the Gibson Bounds over Finite Fields. Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 395-416. http://geodesic.mathdoc.fr/item/SMJ2_2012_38_1-3_a17/