I am revisiting algebraic curves as level sets (equations) or as watersheds (inequations) for geometric and computer graphics purposes.
Now, if I am to evaluate every multivariate polynomial for every pixel in a high resolution image, it will require lots of computations. Do there exist any theoretical results which I can use to create something of a bounding box which I will not need to perform calculations outside of?
In other words, can I find a square or a circle where I can decide that outside of this there will be no values $=0$ or $\leq 0$ ?
for example $$x^4+y^4 - 2^4 = 0$$ will be a shape limited by a square centered on origo with side $4$. How can this be verified?
We can represent this for example with $$\cases{x>2\\x<-2\\y>2\\y<-2}$$
We can calculate gradient $x: 4x^3, y= 4y^3$
But what then? Gradient only gives local information, doesn't it?
Maybe if we consider differential equation with boundary conditions given by the bounding box..?
Do there exist tools estimating range of solutions for differential equations with boundary conditions?