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- W2269618328 abstract "Let sigma be an automorphism of a commutative k-algebra R. The skew polynomial ring R[t;sigma] is generated by R and an indeterminate t subject to the relations ta=sigma(a)t for all a in R. For certain R and appropriate sigma there are elements a and b in R such that the subalgebra of R[t;sigma] generated by at and bt is a free algebra. If sigma is an automorphism of the polynomial ring k[x,y], then the subalgebra of k[x,y][t;sigma] generated by xt and yt is free if and only if sigma is not conjugate to an elementary automorphism. If sigma is an automorphism of k[x,x^{-1},y,y^{- 1}] of the form sigma(x)=x^ay^b and sigma(y)=x^cy^d, then the subalgebra of k[x,x^{-1},y,y^{- 1}][t;sigma] generated by xt and yt is free if the spectral radius of the 2x2 matrix {{a b} {c d}} is >2; indeed, k[x,x^{-1},y,y^{- 1}][t;sigma] contains a free subalgebra if and only if the spectral radius of 2x2 matrix {{a b} {c d}} is >1." @default.
- W2269618328 created "2016-06-24" @default.
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- W2269618328 date "2013-08-07" @default.
- W2269618328 modified "2023-09-27" @default.
- W2269618328 title "Free subalgebras of the skew polynomial rings k[x,y][t;sigma] and k[x,x^{-1},y,y^{-1}][t;sigma]" @default.
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