A gömböc is a $3$-dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c). Such a convex body keeps its properties under tiny perturbations. There exists thus a gömböc given by algebraic equations. What is the simplest (small degree and small coefficients) polynomial $P(x,y,z)\in\mathbb R^3$ such that $P(x,y,z)\leq 0$ defines a gömböc?

Added after Stopple's remarks (1 2): Existence of an algebraic gömböc is not obvious. I should have asked first for existence.

Two other questions concerning these objects are:

Is the set of all (algebraic) gömböcs connected?

Are there any gömböcs in dimension $>3$? (There are none in dimension $2$, every $2$-dimensional convex set has at least $4$ equilibrium points.)