Minimal determinantal representations of bivariate polynomials
Plestenjak, Bor
Za kvadratov prost polinom dveh spremenljivk ▫$p$▫ stopnje ▫$n$▫ je predstavljen enostaven in hiter numerični algoritem za konstrukcijo takih ▫$n \times n$▫ matrik ▫$A$▫, ▫$B$▫ in ▫$C$▫, da velja ...▫$\det(A+xB+yC)=p(x,y)$▫. To je minimalna velikost, ki je potrebna za predstavitev polinoma dveh spremenljivk stopnje ▫$n$▫. V kombinaciji z razcepom na kvadratov proste faktorje je tako možno izračunati ▫$n \times n$▫ matrike za poljuben polinom dveh spremenljivk stopnje ▫$n$▫. Obstoj takih simetričnih matrik je dokazal Dixon že leta 1902, a do sedaj ni bila znana nobena enostavna numerična konstrukcija, tudi če so matrike lahko nesimetrične. Tovrstne upodobitve lahko uporabimo za to, da učinkovito numerično rešimo sistem dveh polinomov dveh spremenljivk nizke stopnje s pomočjo dvoparametričnih problemov lastnih vrednosti. Nova upodobitev znatno pospeši izračun.
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