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f-zpd algebras and a multilinear Nullstellensatz
Bajuk, Žan ...
Let ▫$f=f(x_1,\dots,x_m)$▫ be a multilinear polynomial over a field ▫$F$▫. An ▫$F$▫-algebra ▫$A$▫ is said to be ▫$f$▫-zpd (▫$f$▫-zero product determined) if every ▫$m$▫-linear functional ...▫$\varphi\colon A^{m}\rightarrow F$▫ which preserves zeros of ▫$f$▫ is of the form ▫$\varphi (a_1,\dots,a_m)=\tau(f(a_1,\dots,a_m))$▫ for some linear functional ▫$\tau$▫ on ▫$A$▫. We are primarily interested in the question whether the matrix algebra ▫$M_d(F)$▫ is ▫$f$▫-zpd. While the answer is negative in general, we provide several families of polynomials for which it is positive. We also consider a related problem on the form of a multilinear polynomial ▫$g=g(x_1,\dots,x_m)$▫ with the property that every zero of ▫$f$▫ in ▫$M_d(F)^{m}$▫ is a zero of ▫$g$▫. Under the assumption that ▫$m < 2d-3$▫, we show that ▫$g$▫ and ▫$f$▫ are linearly dependent.
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