20.500.12556/RUL-151007 f-zpd algebre magistrsko delo f-zpd algebras Za multilinearen polinom $f$ stopnje $m$ nad poljem $F$ pravimo, da je algebra $A$ $f$-zpd, če je vsak multilinearen funkcional $\varphi:A^m\rightarrow F$, ki ohranja ničle $f$, oblike $\varphi(x_1,\ldots,x_m) = \tau(f(x_1,\ldots,x_m))$ za vse $x_1,\ldots,x_m$ iz $A$. V delu se ukvarjamo z vprašanjem, katere algebre so $f$-zpd. Posebej osredotočimo na matrične algebre $M_d(F)$. Poleg tega se v delu ukvarjamo s sorodnim problemom, ko multilinearen polinom $g$ ohranja ničle multilinearnega polinoma $f$ za neko algebro. Na koncu si ogledamo aplikacijo $f$-zpd algeber na linearnih preslikavah, ki ohranjajo ničelne produkte, tj. na linearnih preslikavah $T$ med algebrama $A$ in $B$, kjer je $T(x)T(y)=0$, kadarkoli je $xy=0$. For a multilinear polynomial $f$ of degree $m$ over field $F$, we say that an algebra $A$ is $f$-zpd, if every multilinear functional $\varphi:A^m\rightarrow F$ that preserves zeros of $f$ is of form $\varphi(x_1,\ldots,x_m) = \tau(f(x_1,\ldots,x_m))$ for all $x_1,\ldots,x_m$ in $A$. We will be interested in finding $f$-zpd algebras, focusing particularly on matrix algebras $M_d(F)$. Moreover, we will try to answer a similar problem when a multilinear polinomial $g$ is preserving zeros of a multilinear polynomial $f$ for some algebra. Finally we will consider an application of $f$-zpd algebras on linear maps that preserve zero products, ie. on linear maps $T$ between algebras $A$ and $B$ such that $T(x)T(y)=0$ if $xy=0$. algebra določena z ničenim produktom zpd algebra $f$-zpd algebra matrična algebra multilinearni polinom idempotenti Nullstellensatz nič zero product determined algebra zpd algebra $f$-zpd algebra matrix algebra multilinear polynomial idempotent Nullstellensatz zero product true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2023-09-27 08:15:02 2023-09-27 08:15:05 2024-05-29 09:47:46 0000-00-00 00:00:00 2023 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 137063 166778115 10378.pdf 10378.pdf 1 136F380BA59BF8008E9668AF8CEFC951 3e8266aac5a9f1fc59b2ec29f561e671f5388a3dd558e883edd4455e6bd168c2 34656176-5cfd-11ee-a9d7-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=175889 Fakulteta za matematiko in fiziko 0 0 0