Find multiple roots of a three-dim system

Question

Consider the three equations $$ y-x^2=0,\quad z+xy=0,\quad -y-z+x^2-xy+y^2+z^2-x^4=0. $$ How can I find multiple roots of this?

Is it allowed to reduce the system as far as possible and then to find multiple roots of one equation?

For example, we have $$ y=x^2,\quad z=-x^3 $$ from the first two equations, and plugging this into the third, we get $x^6=0$. So $x=0$ has multiplicity 6, and so $(0,0,0)$ has multiplicity 6.

Is thios a general technique?

Answer 1

Look up elimination theory. The example you gave is quite simple, but if you have a complicated system of three equations and three unknowns, one efficient way is via resultants. Given,

$$P_1(x,y,z) = 0\tag1$$

$$P_2(x,y,z) = 0\tag2$$

$$P_3(x,y,z) = 0\tag3$$

Use resultants to eliminate $z$ between $(1),(2)$; then between $(1),(3)$. The command in Mathematica is $Resultant[poly_1,poly_2,z]$. You'll get,

$$R_1(x,y) = 0\tag4$$

$$R_2(x,y) = 0\tag5$$

Then eliminate $y$ between $(4), (5)$ and you'll have an equation solely in $x$.

Answer 2

Using $S$-polynomials for your three given polynomials one can obtain a Groebner basis. This gives the same as elimination, namely only the trivial solution $(x,y,z)=(0,0,0)$.