Find Parameter Set for N Intersections of 2 Implicit Functions

In summary, the speaker is wondering if there are any established numerical techniques to speed up the search for a set of parameters that will result in a specific number of intersections between two implicit functions. The suggested approach is to apply algebraic geometry.
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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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  • #2
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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1. How do you find the parameter set for N intersections of 2 implicit functions?

To find the parameter set for N intersections of 2 implicit functions, you can use a variety of methods such as the elimination method, substitution method, or graphing method. These methods involve manipulating the equations to solve for the variables and then finding the values that satisfy both equations.

2. Can the parameter set vary for different numbers of intersections?

Yes, the parameter set can vary for different numbers of intersections. The number of intersections depends on the number of variables and equations in the system. For example, a system with two equations and two variables may have one or two solutions, while a system with three equations and three variables may have one, two, or three solutions.

3. Are there any limitations to finding the parameter set for N intersections?

There are limitations to finding the parameter set for N intersections, such as the complexity of the equations and the methods used. If the equations are highly nonlinear or have multiple variables, it may be difficult to find the exact parameter set. Additionally, some methods may not work for certain types of equations.

4. How can I verify the accuracy of the parameter set for N intersections?

To verify the accuracy of the parameter set, you can plug the values into the original equations and see if they satisfy both equations. Additionally, you can use a graphing calculator or software to graph the equations and see if the points of intersection match the values in the parameter set.

5. Can the parameter set for N intersections be used to solve real-world problems?

Yes, the parameter set for N intersections can be used to solve real-world problems. Many real-world phenomena can be described by implicit functions, and finding the parameter set can help identify solutions or points of intersection in these systems. This can be useful in fields such as engineering, physics, and economics.

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