VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
  • Noncommutative polynomials nonnegative on a variety intersect a convex set
    Helton, J. William, 1944- ; Klep, Igor, matematik ; Nelson, Christopher
    Let ▫$V \subset \Bbb R^n$▫ be an algebraic set and let ▫${\cal P}(V)$▫ be the ring of polynomial functions on ▫$V$▫. Given ▫$f_1, \dots, f_m \in {\cal P}(V)$▫, define ▫$$ W:= \{x \in V: f_1(x) \geq0, ... \dots, f_m(x) \geq 0\}, $$▫ and let ▫$P_{W}$▫ be the cone of ▫${\cal P}(V)$▫ generated by ▫$f_1, \dots , f_m$▫; that is, the family of those ▫$h \in {\cal P}(V)$▫ that can be written as ▫$h:= s+\sum_{i=1}^rs_ip_i$▫ for some nonnegative integer ▫$r$▫, where ▫$s$▫ and each ▫$s_i$▫ are (finite) sums of squares of elements in ▫${\cal P}(V)$▫ and each ▫$p_i$▫ is a (finite) product of some ▫$f_j$▫'s. The Nichtnegativstellensatz discovered by G. Stengle [Math. Ann. 207, 87--97 (1973)] states that a necessary and sufficient condition for a function ▫$f \in {\cal P}(V)$▫ to be ▫$\geq 0$▫ on each point in ▫$W$▫ is the existence of a nonnegative integer ▫$k$▫ and two functions ▫$g,h \in P_W$▫ such that ▫$fg = f^{2m}+h$▫. The main result of this article is Theorem 1.9, which can be understood as a noncommutative analogue of Stengle's Nichtnegativstellensatz. This very long article (69 pages) contains many other results. Fortunately, its excellent writing makes it nearly as detailed and self-contained as a book, and so it is not only an excellent paper for specialists but it can also be used to introduce young researchers to this exciting topic.
    Vir: Journal of functional analysis. - ISSN 0022-1236 (Vol. 266, iss. 12, 2014, str. 6684-6752)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2014
    Jezik - angleški
    COBISS.SI-ID - 17631833

    Povezava(-e):

    http://dx.doi.org/10.1016/j.jfa.2014.03.016

vir: Journal of functional analysis. - ISSN 0022-1236 (Vol. 266, iss. 12, 2014, str. 6684-6752)
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