Solving polynomial equations

 

• solving linear systems is easy as only the inversion of a matrix is required

• system of polynomial equations

  
  
  

• Grobner basis according to lexicographic ordering of variables (very high time complexity)

• the system of polynomial equations and its Grobner basis have the same solution

• example

  
  
  

• Grobner basis of these polynomials with variables ordering

  
  
  

• this is in "triangular" form

• can be determined by solving the 3rd equation; after substituting into the 2nd equation, we can determine , etc.

• one approximate solution is

• solving (even numerically) of a polynomial equation in one variable is much simpler than solving a system of polynomial equations

• solving a polynomial system is transformed into the successive solution of equations in one variable; see the example Grobner bases