ИСТИНА

The talk is based on the papers \cite{BG,DGK,DGKO}. Two important functions in matrix theory, determinant and permanent, look very similar: $$ \det A= \sum_{\sigma\in { S}_n} (-1)^{\sigma} a_{1\sigma(1)}\cdots a_{n\sigma(n)} \quad \mbox{ and } \quad {\rm per}\, A= \sum_{\sigma\in { S}_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)} $$ here $A=(a_{ij})\in M_n({\mathbb F})$ is an $n\times n$ matrix and ${ S}_n$ denotes the set of all permutations of the set $\{1,\ldots, n\}$. While the computation of the determinant can be done in a polynomial time, it is still an open question, if there are such algorithms to compute the permanent. Due to this reason, starting from the work by P\'olya \cite{P}, 1913, different approaches to change the matrix in such a way that the permanent of original matrix would be equal to the determinant of the new matrix (to convert the permanent into the determinant) were under the intensive investigation. Among our results we prove the following theorem: \begin{Theorem}\label{theorem} Suppose $n\ge 3$, and let ${\mathbb F}$ be a finite field with ${\rm char}\, {\mathbb F}\ne2$. Then, no bijective map $T:M_n({\mathbb F})\to M_n({\mathbb F})$ satisfies $$ {\rm per}\, A=\det T(A). $$ \end{Theorem} Also we investigate Gibson barriers (the maximal and minimal numbers of non-zero elements) for sign-convertible $(0,1)$-matrices and solve several related problems. In particular, we find Gibson barriers for symmetric and weak symmetric conversion. Our results are illustrated by the number of examples. \begin{thebibliography}{99} \bibitem{BG} Budrevich M., Guterman A. Permanent has less zeros than determinant over finite fields // {American Mathematical Society, Contemporary Mathematics.\/} 2012. V. {579}. P. 33-42. \bibitem{DGK} Guterman A., Dolinar G., Kuzma B., P\'olya's convertibility problem for symmetric matrices // {Mathematical Notes.} 2012. V.{ 92},\No 5. P. 684-698. \bibitem{DGKO} Dolinar G., Guterman A., Kuzma B., Orel M. On the P\'olya's permanent problem over finite fields // { European Journal of Combinatorics\/}. 2011. V. 32. P. 116-132. \bibitem{P} P\'olya G. { Aufgabe\/} { 424} // Arch. Math. Phys.1913. V. {20}, \No 3. P. 271. \end{thebibliography}