I am considering three general tori in 3D space, each defined by a quartic equation. My primary question revolves around the number of real solutions that arise from the system of these three equations. The tori are in general positions, meaning they are not constrained to shared symmetries like a common z-axis or concentric arrangements, which are typical simplifications in textbook problems.
From an algebraic standpoint, given that each torus is represented by a quartic equation, one might expect up to $4^3=64$ solutions in total, as a direct application of Bézout's Theorem would suggest. However, the actual number of real solutions is less clear due to the complex interplay between the surfaces. I am interested in understanding the factors that contribute to the number of real solutions and how to determine them.
Motivation: The investigation into the intersection of toroidal shapes has applications in various fields, such as computer graphics, where understanding the interaction of light with such surfaces can be crucial, or in the design of complex engineering structures where toroidal sections intersect. Furthermore, from a mathematical perspective, this question lies at the intersection of algebraic geometry and real analysis.
Current Progress: I have attempted to visualize the problem using graphical software and have derived the quartic equations representing each torus. Nevertheless, the real-world calculation or estimation of their intersections has proven challenging.
Possible Strategies I'm Considering:
Numerical Methods: Using computational software to approximate the intersections, although this does not provide a general solution.
Algebraic Geometry Techniques: Investigating the application of resultants to reduce the problem to a univariate polynomial, whose real roots could be more readily analyzed.
Question for the Community: What strategies might one employ to determine the maximum number of real intersections between these tori? Additionally, is there a known method or result that would help in predicting or bounding the number of real solutions in such a scenario? Any insights, references, or suggestions for further reading would be greatly appreciated.