Suppose F is a field of finite characteristic and u,v,w lie in A=F[x,y]. Let a_n be the dimension of the vector space A/(u^n,v^n,w^n). Suppose a_1 is >0 and finite. Then as n grows, (a_n)/(n^2), though bounded above and bounded away from 0 generally has an oscillating "fractal-like" behavior.
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