We have equations $$a_1x+b_1y+c_1z=d_1$$ $$a_2x^2+b_2y^2+c_2z^2=d_2$$ $$a_3x^3+b_3y^3+c_3z^3=d_3$$

where $a_i,b_i,c_i\in\Bbb N$ at $i\in\{1,2,3\}$ are known.

Is there an efficient ($O(\log(d_1d_2d_3))$ time) procedure to find  $x,y,z\in\Bbb N$ as solutions sought?

I can reduce to $$c_1^2a_2x^2+c_1^2b_2y^2+ c_2(d_1-a_1x-b_1y)^2=c_1^2d_2$$ $$c_1^3a_2x^2+c_1^3b_2y^2+ c_3(d_1-a_1x-b_1y)^3=c_1^3d_3$$

We have the equations

$$a_1x+b_1y+c_1z=d_1,$$ $$a_2x^2+b_2y^2+c_2z^2=d_2,$$ $$a_3x^3+b_3y^3+c_3z^3=d_3,$$

where $a_i,b_i,c_i \in\Bbb N$ at $i \in \{1,2,3\}$ are known.

Is there an efficient ($O(\log(d_1d_2d_3))$ time) procedure to find  $x,y,z\in\Bbb N$ as solutions sought?

I can reduce the system of equations to $$c_1^2a_2x^2+c_1^2b_2y^2+ c_2(d_1-a_1x-b_1y)^2=c_1^2d_2,$$ $$c_1^3a_2x^2+c_1^3b_2y^2+ c_3(d_1-a_1x-b_1y)^3=c_1^3d_3.$$

We have equations $$a_1x+b_1y+c_1z=d_1$$ $$a_2x^2+b_2y^2+c_2z^2=d_2$$ $$a_3x^3+b_3y^3+c_3z^3=d_3$$

where $a_i,b_i,c_i\in\Bbb N$ at $i\in\{1,2,3\}$ are known.

Is there an efficient ($O(\log(d_1d_2d_3))$ time) procedure to find $x,y,z\in\Bbb N$ as solutions sought.

We have equations $$a_1x+b_1y+c_1z=d_1$$ $$a_2x^2+b_2y^2+c_2z^2=d_2$$ $$a_3x^3+b_3y^3+c_3z^3=d_3$$

where $a_i,b_i,c_i\in\Bbb N$ at $i\in\{1,2,3\}$ are known.

Is there an efficient ($O(\log(d_1d_2d_3))$ time) procedure to find $x,y,z\in\Bbb N$ as solutions sought?

I can reduce to $$c_1^2a_2x^2+c_1^2b_2y^2+ c_2(d_1-a_1x-b_1y)^2=c_1^2d_2$$ $$c_1^3a_2x^2+c_1^3b_2y^2+ c_3(d_1-a_1x-b_1y)^3=c_1^3d_3$$