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SUMMARY:Parnashree Ghosh (Indian Statistical Institute\, Kolkata\, India)
DTSTART:20221014T120000Z
DTEND:20221014T130000Z
DTSTAMP:20260423T035541Z
UID:VCAS/141
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VCAS/141/">O
 n the triviality of a family of linear hyperplanes</a>\nby Parnashree Ghos
 h (Indian Statistical Institute\, Kolkata\, India) as part of IIT Bombay V
 irtual Commutative Algebra Seminar\n\n\nAbstract\nLet k be a field\, m a p
 ositive integer\, $\\mathbb{V}$ an affine subvariety of $\\mathbb{A}^{m+3}
 $ defined by a linear relation of the form $x_1^{ r_1}\\cdots x_r^{r_m} y 
 = F(x_1\,\\ldots \, x_m\, z\, t)\,$ A the coordinate ring of $\\mathbb{V}$
  and $G = X_1^{ r_1} \\cdots X_r^{r_m} Y − F(X_1\, \\ldots \, X_m\, Z\, 
 T).$ We exhibit several necessary and sufficient conditions for V to be is
 omorphic $\\mathbb{A}^{m+2}$ and G to be a coordinate in $k[X_1\, \\ldots 
 \, X_m\, Y\, Z\, T]\,$ under a certain hypothesis on F. Our main result im
 mediately yields a family of higher-dimensional linear hyperplanes for whi
 ch the Abhyankar-Sathaye Conjecture holds.\nWe also describe the isomorphi
 sm classes and automorphisms of integral domains of the type A under certa
 in conditions. These results show that for each integer $d\\geq 3\,$ there
  is a family of infinitely many pairwise non-isomorphic rings which are co
 unterexamples to the Zariski Cancellation Problem for dimension d in posit
 ive characteristic. \nThis is a joint work with Neena Gupta.\n
LOCATION:https://researchseminars.org/talk/VCAS/141/
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