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It appears that the only thing that PolynomialReduce guarantees is that p[x,y]= c1 b1[x,y] + c2 b2[x,y] + ... + cn bn[x,y] + r where {{c1,c2,...,cn},r} is the result of polynomial reduce, as it is for your example. Not that the result is in any way minimal or unique. In your example p[x,y] = x^4 - 1, so in fact the 'best' result would be {{0,0,1},0}, but ...

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