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I would like to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with
$x$ and $y$, respectively. How to do it with the simplest code in Mathematica?

For example,

\begin{align*} (A+B)^3-A^3-B^3 &= 3AB(A+B)\\ &= 3xy \end{align*}

In other words, having $A+B=x$ and $AB=y$, how do I express $(A+B)^{10}-A^{10}-B^{10}$ in terms of $x$ and $y$.

Bonus question

Rather than creating a new question for it, I think I should ask here. 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?

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You can use GroebnerBasis:

eq = (a + b)^10 - a^10 - b^10;
eqXY = GroebnerBasis[{eq, a + b - x, a b - y}, {x, y}, {a, b}];

(*out*){10 x^8 y - 35 x^6 y^2 + 50 x^4 y^3 - 25 x^2 y^4 + 2 y^5}

Check:

First@Expand[eqXY /. x -> (a + b) /. y -> a b] === Expand[eq]
(*out*)True

--EDIT--

Following @DanielLichtblau's suggestion, it's better to do this in two steps: first find a Groebner basis (of the polynomials corresponding to the transformation equations) and then reduce your polynomial in terms of the basis to account for the case where not all variables get eliminated. I trust he knows what he's talking about having written parts of the function :). So,

With[{
   vars = {a, b},
   relations = {a + b - x, a b - y},
   poly = (a + b)^10 - a^10 - b^10},
  PolynomialReduce[poly, GroebnerBasis[relations, vars], 
   vars]] // Last
(*out*){10 x^8 y - 35 x^6 y^2 + 50 x^4 y^3 - 25 x^2 y^4 + 2 y^5}

is safer. In fact, I realised your question is answered by one of the examples in the "Applications" section of the documentation for PolynomialReduce

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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? $\endgroup$ – kiss my armpit Oct 4 '13 at 13:38
  • $\begingroup$ GroebnerBasis[{a^10 + b^10, a + b - x, a b - y}, {x, y}, {a, b}] produces the expected result. $\endgroup$ – gpap Oct 4 '13 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. $\endgroup$ – Daniel Lichtblau Oct 4 '13 at 14:06
  • $\begingroup$ Thanks a lot - I edited to match the example in the documentation. $\endgroup$ – gpap Oct 4 '13 at 18:24
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You can use Simplify and give your replacements as assumptions:

Simplify[(a + b)^3 - a^3 - b^3, a + b == x && a b == y]
(* 3 x y *)

(* For higher n seems like you have to Expand first *)
Simplify[(a + b)^10 - a^10 - b^10 // Expand, a + b == x && a b == y]
(* y (10 x^8 - 35 x^6 y + 50 x^4 y^2 - 25 x^2 y^3 + 2 y^4) *)

(* Or alternatively use a custom ComplexityFunction *)
Simplify[
 (a + b)^n - a^n - b^n, a + b == x && a b == y,
 ComplexityFunction -> (Count[#, a | b, Infinity] &)] // Simplify
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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? $\endgroup$ – kiss my armpit Oct 4 '13 at 13:34
  • $\begingroup$ @PGFTricks The goal of Simplify is to minimize its ComplexityFunction the default one is similar to LeafCount and a^10 + b^10 is pretty darn simple as far as it is concerned. You can search ComplexityFunction on this site for examples with adding penalty to expressions you don't want when simplifying. $\endgroup$ – ssch Oct 4 '13 at 13:36
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A very simple approach is the following :

el[n_] := Eliminate[ {r == (A + B)^n - A^n - B^n , x == A + B , y == A*B}, {A,B}]//First

Expand[Array[el, 10]] // TableForm
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An approach:

pol[n_] := Expand[(a + b)^n - a^n - b^n]
sol = Solve[{a + b == x, a b == y}, {a, b}][[1]];
fun[n_] := Expand@Simplify[pol[n] /. sol]

Tabulating:

Table[{(a + b)^j - a^j - b^j, fun[j]}, {j, 2, 10}] // Grid

enter image description here

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