Tarski-Seidenberg theorem

Tarski-Seidenberg theorem

States that a set of equalities and inequalities f(x1,x2,x3,....xn)>=0 can be reduced down to a set of equalities and inequalities f(x1,x2,x3,....xn-1)>=0 with one less variable.

Thus all equations can be reduced to a set of equalities and inequalities on their coefficients.

See also: cylindrical algebraic decomposition

 See: Elimination

These algorithms aren't very simple.

Comments

Post a Comment

Popular posts from this blog

Vortex Solution to Navier Stokes Equations

Image
This is a solution I found to the Navier Stokes equations . It uses the flow lines of the Hopf Fibration (which is basically just a way of covering the entire 3D space in circles). The flow lines can be seen in this image below: You can see my paper here. Solution to Navier-Stokes equation . Unfortunately I won't get a million pounds for winning the Clay Millennium prize because this is a compressible not incompressible solution. :( Oh, well. Nice try! I think it's pretty nice anyway. :) Also it uses the nice identity: which paramaterizes the Pythagorean quadruples!
Read more

Solving two polynomials in two variables

Solving two polynomials in two variables I had an algorithm for solving two polynomials in 2 variables by converting it into an equation in 1 variable. I'll see if I can remember it. Let us start with 2 polynomials. (solution: x=2 y=3). We try to eliminate x. (1) x 2 +4xy+2y 2 +3x+4y - 64 = 0 (2) 3x 2  -xy + y 2 +5x-y -22 = 0 First eliminate x 2 (3) =- (2) +3 (1) :  13xy +5y 2  +4x +13y  -170 = 0 or (13y+4)x + (5y 2 +13y  -170) = 0 Next use this equation to eliminate x 2  again from (1) . (1) .(13y+4) :  (13y+4)x 2 + (52y 2 +55y+12)x + (13y+4)(2y 2 +4y - 64) = 0 - (3) .x:             (13y+4)x 2  + (5y 2 +13y  -170)x = 0 =(4) : (47y 2 +42y+182)x + (13y+4)(2y 2 +4y - 64) = 0 Now use these two linear equations to eliminate x: (13y+4) (4) -  (47y 2 +42y+182) (3)  :   (13y+4)(13y+4)(2y 2 +4y - 64) - (47y 2 +42y+182)(5y 2 +13y  -170) =0 or  (y-3) ...
Read more