Existence of non-trivial solution to non linear polynomial system

Question

I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases:

The first case:

$f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$

$f2: b_1x^2+b_2xy+b_3y^2+b_4z^2=0$

The second case:

$f3: a_1x^2+a_2xy+a_3y^2+a_3z^2+a_5xz=0$

$f4: b_1x^2+b_2xy+b_3y^2+b_3z^2+b_5xz=0$

(note that the coefficient of y2 and z2 are equal)

All the coefficient are reals number and the variable x,y,z should be real too.

I tried to use resultant and eliminating one variable, for even a more simple case:

$f5: a_1x^2+a_2xy+a_3y^2+a_3z^2=0$

$f6: b_1x^2+b_2xy+b_3y^2+b_3z^2=0$

The problem that i get different solution:

1: if I try to eliminate $z$, I get $RES(f5,f6,z)=x^2[(a_2c_1−a_1c_2)x+2(b_2c_1−b_1c_2)y]^2$ So there is always non trivial (x≠0 or y≠0) real x,y such that RES(f5,f6,z)=0

2: On the other hand if I eliminte $x$ I get: $ \small RES(f5,f6,x)=[(a_2c_1−a_1c_2)^2+4(a_2b_1−a_1b_2)(b_2c_1−b_1c_2)]x^4+4(b_2c_1−b_1c_2)^2x^2z^2$

thus a non-trivial solution exist only if $(a_2c_1−a_1c_2)^2+4(a_2b_1−a_1b_2)(b_2c_1−b_1c_2)≤0$.

I tried to check this numerically and it seems like the second condition is the right one, so why the first attempt eliminating z is wrong?

Any help will be appreciated

Answer 1

Using Groebner bases (or even direct computing) we see, that the solutions of your system $f_5=f_6=0$ are as follows:

Case 1: $a_1b_2-a_2b_1=0$. Then it follows $y^2+z^2=0$. Over the real numbers this means $y=z=0$, and $a_1x=b_1x=0$. For $a_1=b_1=0$ we have always a solution with $x\neq 0$.

Case 2: $a_1b_2-a_2b_1\neq 0$. Then we obtain either $y=0$ and $x=z=0$, or if $y\neq 0$, then $$ x=\frac{(a_3b_1-a_1b_3)(y^2+z^2)}{(a_1b_2-a_2b_1)y}, $$

and $ry^2+sz^2=0$ with $$ r:= a_1^2b_3^2 - a_1a_2b_2b_3 - 2a_1a_3b_1b_3 + a_1a_3b_2^2 + a_2^2b_1b_3 - a_2a_3b_1b_2 + a_3^2b_1^2 $$ and $$ s:= (a_1b_3 - a_3b_1)^2. $$ Here you have the case 2, i.e., either $y=z=0$ again (we have required $y\neq 0$, so this was solved earlier), or some non-trivial solution. The other systems can be solved too, using Groebner bases.