Solving a system of univariate polynomials

Question

How would one go about solving a system of homogeneous univariate polynomials. For instance,

$$a_1x^4+ ax^3 + bx^2 + cx = 0 $$ $$d_1x^4 +dx^3 + ex^2 + fx= 0 $$ $$g_1x^4 +gx^3+hx^2 + ix = 0$$ $$l_1x^4 + lx^3+ mx^2+ nx =0. $$ I am interested in non-trivial solutions over $\mathbb{C}.$

Answer 1

$x=0$ is an obvious solution. To look for others, divide the equations by $x$ then eliminate $x^2$ between the first two:

$$0=d(ax^2+bx+c)-a(dx^2+ex+f)=(bd-ae)x+(cd-af)$$

• If $bd-ae \ne 0$ then the above implies $x=\frac{af-cd}{bd-ae}$. Substitute this solution back into the $3$ original equations. If it satisfies them, then that's the unique non-$0$ solution. Otherwise there are no non-$0$ solutions.

• If $bd-ae = 0$ then if $cd-af \ne 0$ there are no no non-$0$ solutions. Otherwise if $cd-af=0$ the first two equations are identical but for a multiplicative factor. Drop one of them, then repeat the steps above for the remaining two equations.

Answer 2

A general method for finding the solutions of a system of simultaneous polynomial equations $$ f_1(x)=0, \ \ f_2(x)=0, \ \ ... \ \ f_n(x)=0 $$ is finding the $\gcd\big(f_1, f_2, ..., f_n\big)=g(x)$, of these polynomials.

Recall at this point that the ring $\mathbb{C}[x]$, i.e. the ring of polynomials with complex coefficients, is a PID (a principal ideal domain). This means that any of its ideals is a principal ideal, i.e. an ideal generated by a single element. In particular, regarding the ideal of $\mathbb{C}[x]$ generated by the polynomials $f_1, f_2, ..., f_n$, it can be shown that it is generated by their greatest common divisor: $$ <f_1, f_2, ..., f_n>=<\gcd\big(f_1, f_2, ..., f_n\big)>=<g> $$ Thus, the solution of the initial system, is equivalent to the solution of the single equation $$ g(x)=0 $$ P.S.: The $\gcd\big(f_1, f_2, ..., f_n\big)$, can be found either by straightforward factorization of the polynomials $f_1, f_2, ..., f_n$, or more generally using the Euclidean division algorithm for finding the $\gcd$ of two univariate polynomials in $\mathbb{C}[x]$ together with the fact that for $n\geq 3$ $$ \gcd(f_1, f_2, ..., f_n)=\gcd\big(f_1,\gcd(f_2, ..., f_n)\big) $$