Find Parameter Set for N Intersections of 2 Implicit Functions

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SUMMARY

This discussion focuses on finding parameter sets for two implicit functions, F(x,y)=0 and G(x,y)=0, to determine their intersections in the x-y plane. The functions are typically high-degree rational polynomials dependent on a K-dimensional parameter set. Established numerical techniques, particularly those rooted in algebraic geometry, are suggested to expedite the search for parameters yielding a specified number of intersections, N. The conversation emphasizes the complexity of the problem due to the high dimensionality of the parameter space.

PREREQUISITES
  • Understanding of implicit functions and their properties
  • Familiarity with high-degree polynomial equations
  • Knowledge of algebraic geometry concepts
  • Experience with numerical methods for parameter optimization
NEXT STEPS
  • Research algebraic geometry techniques for analyzing implicit functions
  • Explore numerical optimization methods for high-dimensional parameter spaces
  • Learn about polynomial root-finding algorithms
  • Investigate software tools for symbolic computation, such as Mathematica or Maple
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Mathematicians, data scientists, and engineers working with complex systems involving implicit functions and seeking to optimize parameter searches for intersection analysis.

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Sorry for the wordiness of the thread title.

Basically I'm wondering, if you have two implicit functions, F(x,y)=0 and G(x,y)=0 (typically rational functions with numerator and denominator very high degree polynomials), both dependent upon the same K (in my case > 34) dimensional set of parameters - a for i = 1 to K; are there any established numerical techniques to speed up a search of the parameter space to find a set of parameters for which the 2 curves in the x-y plane (defined by the functions F and G) have a given number of intersections, say N?

Cheers!
 
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You can apply algebraic geometry to some extent, but this depends on many things you haven't said. But algebraic geometry would be the direction in which I sought.
 
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