# Algebraic solution for a system of algebraic equations?

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.

• 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. Oct 1, 2019 at 6:35
• @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?
– Hans
Oct 1, 2019 at 6:56
• 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 Oct 1, 2019 at 8:09
• @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.
– Hans
Oct 1, 2019 at 9:20
– Hans
Oct 1, 2019 at 20:40

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.

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