Given a multivariate polynomial $p(\mathbf x)=p(x_1,...,x_n)$ for $\mathbf x\in\Bbb R^n$. Are there some easy conditions to be set on (the coefficients of) $p$ to ensure that
$$C_p:=\{(x_1,...,x_n)\in\Bbb R^n\mid p(x_1,...,x_n)\le 0\}\subseteq\Bbb R^n$$
is convex? I came across this problem in a lesson on computational algebraic geometry at some point, but I never got an answer or found one myself.
I don't think there's really much in the way of "easy" solutions to this problem (though knowing Cunningham's law, as soon as I post this, someone will show up with a nice method). For instance, $p$ convex certainly implies this result (though certainly there are $p$ which aren't convex but do satisfy this property), and convexity is implied by (though, again, certainly not equivalent to) the Hessian matrix being positive semidefinite everywhere. Checking convexity is algorithmically difficult: it is NP-hard.
On the other hand, there are theorems from the area of semi-algebraic geometry which will handle this for you. See for instance Lasserre, J. B. (2010). Certificates of convexity for basic semi-algebraic sets. Applied Mathematics Letters, 23(8), 912–916. https://doi.org/10.1016/j.aml.2010.04.009 .