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How would one solve algebraically the following system of algebraic equations? $$f(a,b):=a(1-b)+ab\frac a{a+b}.$$ $$u = f(a,b),\quad v = f(b,a).$$ Solve algebraically $(a,b)$ in terms of $(u,v)$


Multiplying both sides of the equations by $a+b$ would give us a system of cubic equations. But that does not seem to help much in solving the equations.

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  • $\begingroup$ You can use the Grobner bases approach, to set up a system of equations, and eliminate a and b from the system: f[a_, b_] := a (1 - b) (a + b) + a b a; GroebnerBasis[{u (a + b) - f[a, b], v (a + b) - f[b, a]}, {a, b}, {u, v}] This does not give any output, so I interpret this as what you are asking for is not possible. $\endgroup$ Commented Oct 1, 2019 at 6:35
  • $\begingroup$ @PerAlexandersson: I am not familiar with Gröbner basis. Could you please describe the approach in details and formulate an answer? Also your $f(a,b)$ is my $f(a,b)$ multiplied by $a+b$, right? How do you reach your conclusion of impossibility exactly? $\endgroup$
    – Hans
    Commented Oct 1, 2019 at 6:56
  • $\begingroup$ Right, I cleared the denominators, since Grobner bases deals with polynomials, see en.wikipedia.org/wiki/Gr%C3%B6bner_basis for info. A standard reference is springer.com/gp/book/9783319167206 $\endgroup$ Commented Oct 1, 2019 at 8:09
  • $\begingroup$ @PerAlexandersson: Thank you very much for the references. However, could you please write out your derivation of the unsolvability in details as a formal answer below? I would like to see the Gröbner basis in action. It will be greatly appreciated. $\endgroup$
    – Hans
    Commented Oct 1, 2019 at 9:20
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    $\begingroup$ @PerAlexandersson: Are you talking about this section en.wikipedia.org/wiki/…? This is the way to deal with rational as opposed to polynomial equations? $\endgroup$
    – Hans
    Commented Oct 1, 2019 at 20:40

1 Answer 1

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This is mechanized in current CASes, e.g. the command of Maple 2019.1

solve({a*(1 - b) + a*b*a/(a + b) = u, eval(a*(1 - b) + a*b*a/(a + b), {a = b, b = a}) = v}, {a, b}, explicit);

performs a long output which can be seen here exported as a PDF file.

Addition. The command of Mathematica

Reduce[{a*(1 - b) + a*b*a/(a + b) == u, b*(1 - a) + b^2*a/(a + b) == v}, {a, b}]//ToRadicals

produces

(u == 1 - v && a == 1 && (b == 1/2 (v - Sqrt[v] Sqrt[4 + v]) || b == 1/2 (v + Sqrt[v] Sqrt[4 + v]))) || ((a == (2 + u)/4 - 1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))) - 1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - v - (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^( 1/3)) - (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))))) || a == (2 + u)/4 - 1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))) + 1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - v - (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^( 1/3)) - (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))))) || a == (2 + u)/4 + 1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))) - 1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - v - (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^( 1/3)) + (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))))) || a == (2 + u)/4 + 1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3))) + 1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) - v - (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^( 1/3)) + (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) - 8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u - v + 1/3 (2 u + v) + (2^( 1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/( 3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) - 72 (2 u + v) (-u^2 - u v) - 9 (-2 - u) (2 u + v) (u + v - u v - v^2) + 27 (u + v - u v - v^2)^2)^2))^(1/3)))))) && -1 + a != 0 && b == (a - u - v)/(-1 + a) && a^2 - u - v != 0)

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1
  • $\begingroup$ Thank you. So there exists an algebraic (radical) solution contrary to what Per Alexandersson claims? Does the difference in claim coming from perhaps Maple and Mathematica use different methods than the Gröbner basis or the two are more powerful than the software Per Alexandersson is using? $\endgroup$
    – Hans
    Commented Oct 1, 2019 at 19:25

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