20.500.12556/RUP-7737 Permanent versus determinant over a finite field Permanente v primerjavi z detminantami nad končnimi polji Let ▫$\mathbb{F}$▫ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices ▫$\mathcal{H}_n(\mathbb{F})$▫ and for the whole matrix space ▫$M_n(\mathbb{F})$▫. It is known that for ▫$n = 2$▫, there are bijective linear maps ▫$\Phi$▫ on ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫ satisfying the condition per ▫$A = \det \Phi(A)$▫. As an application of the obtained results, we show that if ▫$n \ge 3$▫, then the situation is completely different and already for ▫$n = 3$▫, there is no pair ofmaps ▫$(\Phi, \phi)$▫, where ▫$\Phi$▫ is an arbitrary bijective map on matrices and ▫$\phi \colon \mathbb{F} \to \mathbb{F}$▫ is an arbitrary map such that per ▫$A = \phi(\det \Phi(A))$▫ for all matrices ▫$A$▫ from the spaces ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫, respectively. Moreover, for the space ▫$M_n(\mathbb{F})$▫, we show that such a pair of transformations does not exist also for an arbitrary ▫$n > 3$▫ if the field ▫$\mathbb{F}$▫ contains sufficiently many elements (depending on ▫$n$▫). Our results are illustrated by a number of examples. Naj bo ▫$\mathbb{F}$▫ končno polje, katerega karakteristika ni enaka 2. V članku obravnavamo kardinalnost množice matrik, ki imajo enako predpisano vrednost determinante, in kardinalnost množice matrik, ki imajo enako predpisano vrednost permanente. Pri tem se najprej omejimo na hermitske matrike ▫$\mathcal{H}_n(\mathbb{F})$▫, nato pa obravnavamo problem še v okviru množice vseh matrik ▫$M_n(\mathbb{F})$▫. Znano je, da za ▫$n = 2$▫ obstajajo take bijektivne linearne preslikave ▫$\Phi$▫ na ▫$\mathcal{H}_n(\mathbb{F})$▫ in ▫$M_n(\mathbb{F})$▫, da velja ▫$\text{per} A = \det \Phi(A)$▫. S pomočjo dobljenih rezultatov pokažemo, da je za ▫$n \ge 3$▫ situacija povsem drugačna. Že za ▫$n = 3$▫ ne obstaja noben par preslikav ▫$(\Phi, \phi)$▫, kjer bi bila ▫$\Phi$▫ poljubna bijektivna preslikava na matrikah in bi bila ▫$\phi \colon \mathbb{F} \to \mathbb{F}$▫ poljubna preslikava, tako da bi veljalo ▫$\text{per} A = \phi(\det \Phi(A))$▫ za vse matrike ▫$A$▫ iz prostora ▫$\mathcal{H}_n(\mathbb{F})$▫ ali ▫$M_n(\mathbb{F})$▫. Še več, za prostor ▫$M_n(\mathbb{F})$▫ pokažemo, da tak par preslikav ne obstaja za poljuben ▫$n > 3$▫, če polje ▫$\mathbb{F}$▫ vsebuje dovolj elementov (v odvisnosti od ▫$n$▫). Navedenih je tudi več primerov, ki ilustrirajo naše rezultate. mathematics linear algebra matrix theory permanent determinant matematika linearna algebra teorija matrik permanenta determinanta true true false Angleški jezik Slovenski jezik Delo ni kategorizirano 2016-04-08 16:47:17 2017-04-03 08:11:26 2024-03-01 13:33:26 0000-00-00 00:00:00 2013 0 0 Str. 404-413 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 1072-3374 512.643 16715609 16715865 http://dx.doi.org/10.1007/s10958-013-1469-4 1 https://repozitorij.upr.si/Dokument.php?lang=slv&id=7737 Inštitut Andrej Marušič 0 0 0