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I was playing around with gröbner basis and s-polynomials and I fell upon the PolynomialReduce, and I was wondering why it gives different results when I move around the polynomials in its second parameter,

PolynomialReduce[x^2 (x^2 + y) - 1 (x^2 y + 1), {x^2 y + 1, y + x^2, x^4 - 1}, {x, y}]

gives

{{-1, x^2, 0}, 0}

while

PolynomialReduce[ x^2 (x^2 + y) - 1 (x^2 y + 1), {y + x^2, x^2 y + 1, x^4 - 1}, {x, y}]

gives

{{x^2 - y, 0, 0}, -1 + y^2}

which is obviously wrong** as it is very apparent that the coefficients is just x^2 and -1.

Can anybody shed some light on this? It's very disturbing and confusing.

[edit] **the result might not be wrong, but it does not give the minimal b as per instruced by the documentation http://reference.wolfram.com/language/ref/PolynomialReduce.html

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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 this is only obtained when x^4 -1 is the first polynomial.

I am quite sure that result of general polynomial division is not unique unless over a (reduced) Groebner basis. The Groebner basis for the ideal generated by your two polynomial sets are, of course, the same.

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  • $\begingroup$ (John I'm sure knows this but...) just to chime in, this response is correct in every detail. $\endgroup$ – Daniel Lichtblau Aug 5 '14 at 14:18
  • $\begingroup$ An additional though: in the linear algebra case if v, u1, u2 are vectors in Rn then projecting v onto u1, then projecting (v - proj(v,u1)) onto u2 will also give a different result than projecting onto u2 first, unless u1,u2 are orthogonal. $\endgroup$ – John McGee Aug 5 '14 at 14:27
  • $\begingroup$ But the specification reference.wolfram.com/language/ref/PolynomialReduce.html Actually guarantees that 'b is minimal' ? $\endgroup$ – DrLime2k10 Aug 5 '14 at 14:59
  • $\begingroup$ Yes, for a suitable definition of "minimal". It simply means that b cannot be further reduced by the list of polynomials. For "canonical" you need to first create a Groebner basis, per this response. $\endgroup$ – Daniel Lichtblau Aug 5 '14 at 15:39
  • $\begingroup$ Can you explain why I am getting the following results then when I am reducing it to a groebner basis? gb := GroebnerBasis[{y + x^2, x^2 y + 1, x^4 - 1}, {y, x}] PolynomialReduce[x^2 (x^2 + y) - 1 (x^2 y + 1), gb, {y, x}] (*gives result*) {{1, 0}, 0} Now these are just not correct in any sense. $\endgroup$ – DrLime2k10 Aug 5 '14 at 18:12

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