I am recently approaching combinatorial commutative algebra and I am studing Upper Bound Theorem for Simplicial Spheres (Stanley 1975). My question is so a bit general and maybe ingenuous...
Momentum curve is the algebraic curve $M \subset \mathbb{R}^d$ defined parametrically by $x(\tau) = (\tau, \tau^2, ..., \tau^d) $. A cyclic polytope $C(n, d)$ is the convex hull of any $n$ distinct points on $M$.
There is a first Proposition I have seen,that says any cyclic polytope is a symplicial $d$- polytope (since you can prove any $d+1$ dinstinct points on $M$ are affinely independent).
Question 1) Are there other algebraic curves with this property?
More over you can prove the combinatorial type of a cyclic polytope $C(n, d)$ depends only upon $n$ and $d$ and not on the particular vertex set $V \subset M$.
Question 2) Are there other algebraic curves with this property?
So the point is: I do not understand the peculiarity of momentum curve. Maybe there is another curve $N$ such that if you take the convex hull of $n$ distinct points, call it $N(n, d)$, you can reformulate upper bound theorem for simplicial spheres in this way:
Given $P$ a simplicial complex whose geometric realization is topologically a sphere, with n vertices, the following inequality holds: $f_j(P) \le f_j(N(n, d))$, where $f_j$ is any component of the $f$-vector.
If such a curve does not exist, why? I think McMullen's formulated the conjecture, so how did he find the momentum curve?
Thanks
To your last (unnumbered) questions:
Suppose that such a function $N(n,d)$ exists with $f_i(P)\le f_i(N(n,d))$ for all simplicial spheres $P$. This must in particular hold for the boundary complex of the cyclic polytope. Therefore
$$f_i(C(n,d))\le f_i(N(n,d)).$$
But by the upper bound theorem for simplicial spheres (proven by Richard Stanley), we also have
$$f_i(N(n,d))\le f_i(C(n,d))$$
as well, establishing identity, and so $N(n,d)$ must be cyclic.