# Symmetric Systems

Polynomial P(x,y) is called symmetric if interchanging of x and y doesn't change polynomial, i.e. P(x,y)=P(y,x).

For example, polynomial P_1(x,y)=x^3+3xy+y^3 is symmetric because P_1(y,x)=y^3+3yx+x^3=x^3+3xy+y^3=P_1(x,y), and P_2(x,y)=x^2+y is not because P_2(y,x)=y^2+x !=P_2(x,y).

System of equations is called symmetric if both of its equations are symmetric.

Symmetric system can be solved by introducing new variables - standard symmetric polynomials, i.e. x+y and xy.

Example. Solve System of equations {(x^3+x^3y^3+y^3=17),(x+xy+y=5):}.

This is symmetric system. Let u=x+y,\ v=xy. Since x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)((x+y)^2-3xy)=u(u^2-3v)=

=u^3-3uv then given system can be rewritten as {(u^3-3uv+v^3=17),(u+v=5):}.

This system can be easily solved by substitution method (expressing u from second equation and plugging in first). Solutions of this system are (3;2) and (2;3).

When we return to old variables, we will obtain set of systems: {(x+y=3),(xy=2):},{(x+y=2),(xy=3):}.

These systems again can be solved using substitution method. First system has two solutions: (1;2) and (2;1), second system doesn't have solutions.

So, (1;2) and (2;1) are solutions of the initial system.