 # Polynomial Reduction

The applet on this page uses Sylvester's resultants to reduce systems of polynomial equations. For example, it can be used to reduce a system of 4 polynomials in 4 variables to a single polynomial in one of the variables. The reduction can be done all at once, or step by step.

The applet uses code from ring.perisic.com

See below for some examples of how it could be used.

### Examples

Equation of a plane

Want to know the equation of a plane through three points? Suppose I want the equation of the plane through (2,3,4), (-1,2,8) and (4,0,-6). Into the 'enter polynomials' part of the applet, I could type in the equation of the plane `a*x+b*y+c*z-d`. Then, I substitute the three points and type these equations in too (before the equation in x,y and z. `2*x+3*y+4*z-d`, `-a+2*b+8*c-d` and `4*a+0*b-6*c-d`. Into the 'variables' field, I enter the variables : `x,y,z,a,b,c,d`. For this example, it's important that `x,y,z` come before a,b,c,d. This is because we want the equation of the plane in terms of x, y and z.

So, my 'polynomials' section reads `2*a+3*b+4*c-d,-a+2*b+8*c-d,4*a+0*b-6*c-d,a*x+b*y+c*z-d`, and the 'variables' section reads `x,y,z,a,b,c,d`. Clicking 'Fully Reduce' gives `(-88 + 88*x + -88*y + 44*z)*a`, which I can divide by 44a to give the equation `-2+2x-2y+z=0`.

### Finding a Differential Equation

Trying to work out the relationship between the mass and velocity of a rocket with exhaust velocity u is tough. Using conservation of momentum in special relativity gives an awful equation :

mv/sqrt(1-(v/c)2) = (m-dm)(v+dv)/sqrt(1-((v+dv)/c)2) + dm(v-(1-(v/c)2)u)/sqrt(1-(v/c-(1-(v/c)2)u/c)2)!
.
Then, we have to rearrange this to get dv/dm in terms of the other quantities. Multiplying through by c helps a little, but not much.

Then I though, why not write this whole thing as a bunch of polynomials to solve, and let the applet do the donkey work? I typed
`(-m*v/a+(m*v-v*dm+m*dv)/b + dm*x/e)*a*b*e, a^2-c^2+v^2, b^2+(v+dv)^2-c^2,e^2-c^2+x^2, c^2*x-c^2*v-(c^2-v^2)*u, dvdm*dm-dv`
into the 'polynomials' section. I hope you'll agree this is equivalent to the equation above. For the variables, I tried `c,m,v,u,dvdm,dm,a,b,e,x,dv`. Eliminate x first, then e, then b, then a, then dv. This failed completely.