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On a problem of A. V. Grishin | | C. Bekh-Ochir
; S. A. Rankin
; | | Date: |
11 Sep 2009 | | Abstract: | In this note, we offer a short proof of V. V. Shchigolev’s result that over
any field k of characteristic p>2, the T-space generated by
x_1^p,x_1^px_2^p,... is finitely based, which answered a question raised by A.
V. Grishin. More precisely, we prove that for any field of any positive
characteristic, R_2^{(d)}=R_3^{(d)} for every positive integer d, and that over
an infinite field of characteristic p>2, L_2=L_3. Moreover, if the
characteristic of k does not divide d, we prove that R_1^{(d)} is an ideal of
k_0<X> and thus in particular, R_1^{(d)}=R_2^{(d)}. Finally, we show that for
any field of characteristic 3 or 5, R_1^{(3)}
e R_2^{(3)}. For a field of
characteristic p>5, it is not known whether R_1^{(p)}=R_2^{(p)}, nor is it
known whether L_1=L_2 for pge 3. | | Source: | arXiv, 0909.2266 | | Services: | Forum | Review | PDF | Favorites | |
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