Fix a field k of characteristic zero. If a(1), ..., a(n) (n >= 3) are positive integers, the integral domain B-a1,B- (....,) (an) = k [X-1, ..., X-n]/ < X-1(a1) +...+ X-n(an)> is called a Pham-Brieskorn ring. It is conjectured that if a(i) >= 2 for all i and a(i) = 2 for at most one i, then B-a1,B- (...,) (an) is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D : B -> B is the zero derivation.) The conjecture is known to be true when n = 3, and in certain special cases when n >= 4. This article settles several cases not covered by previous results. For instance, we show that if a >= n >= 4 then B-a,B- ...,B- a is rigid (where 'a' occurs n times), and that if Sigma(n)(i=1) 1/a(i) <= 1/n-2 then B-a1,B- ...,B- an is stably rigid.
On the rigidity of certain Pham-Brieskorn rings
Michael Chitayat;
2022
Abstract
Fix a field k of characteristic zero. If a(1), ..., a(n) (n >= 3) are positive integers, the integral domain B-a1,B- (....,) (an) = k [X-1, ..., X-n]/ < X-1(a1) +...+ X-n(an)> is called a Pham-Brieskorn ring. It is conjectured that if a(i) >= 2 for all i and a(i) = 2 for at most one i, then B-a1,B- (...,) (an) is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D : B -> B is the zero derivation.) The conjecture is known to be true when n = 3, and in certain special cases when n >= 4. This article settles several cases not covered by previous results. For instance, we show that if a >= n >= 4 then B-a,B- ...,B- a is rigid (where 'a' occurs n times), and that if Sigma(n)(i=1) 1/a(i) <= 1/n-2 then B-a1,B- ...,B- an is stably rigid.
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