I need to find an analytical solution to a system of bivariate polynomials.


\begin{eqnarray} a_0 + a_1 x + a_2 y + a_3 xy+a_4 x^2 + a_5 y^2 + a_6 xy^2 + a_7 x^2 y + a_8 x^2 y^2 &= 0 \\ b_0 + b_1 x + b_2 y + b_3 xy+b_4 x^2 + b_5 y^2 + b_6 xy^2 + b_7 x^2 y + b_8 x^2 y^2 &= 0 \end{eqnarray}

Thus, the total degree of each monomial is at most 4, but on the individual variables the degree is at most 2.

I understand that systems of polynomial equations are the object of study of algebraic geometry, but to my detriment I am not at all acquainted with this particular branch of mathematics.

How should one go about solving such systems?

Is there a book or lecture notes I could read on this topic?

Searching around I found the books "Ideals, Varieties and Algorithms" by Cox, Little, and O'Shea, and "Solving Systems of Polynomial Equations" by Bernd Strumfels (haven't ordered either of them yet). The problem is that I am not sure how appropriate is the first book for someone someone not too familiar with abstract algebra, and the second book seems oriented towards numerical solutions.

DISCLOSURE: The coefficients do have some structure (not random real numbers) but they are too complex and unwidely to describe in this post. From the geometry of the problem I know that except for a degenerate choice of the parameters of the problem, this system will have two real solutions.


The word to look up is "resultant".

  • $\begingroup$ Short answer, good answer (+1). $\endgroup$ – vesszabo Oct 19 '12 at 20:30
  • $\begingroup$ Thank you Robert, that was all I needed (tried to vote up your answer, but I couldn't). $\endgroup$ – Rick Deckard Oct 22 '12 at 15:09

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