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  • $\begingroup$ If I want to find $a^{10}+b^{10}$ in terms of $x$ and $y$, why does Simplify[(a + b)^10 - Expand[(a + b)^10 - a^10 - b^10], a b == x && a + b == y] not produce the expected result? $\endgroup$ Commented Oct 4, 2013 at 13:38
  • $\begingroup$ GroebnerBasis[{a^10 + b^10, a + b - x, a b - y}, {x, y}, {a, b}] produces the expected result. $\endgroup$ Commented Oct 4, 2013 at 13:42
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    $\begingroup$ This is reasonable (I upvoted..) I'd recommend separating into a GB followed by polynomial reduction though. In[607]:= vars = {a, b, x, y}; In[608]:= PolynomialReduce[(a + b)^3 - (a^3 + b^3), GroebnerBasis[{a + b - x, a*b - y}, vars], vars][[2]] Out[608]= 3 x y Reason being that the replacement will not in general result in certain variables being eliminated, so a GB alone might not suffice. $\endgroup$ Commented Oct 4, 2013 at 14:06
  • $\begingroup$ Thanks a lot - I edited to match the example in the documentation. $\endgroup$ Commented Oct 4, 2013 at 18:24