On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

Markov’s Inequality and $$C^{\infty }$$ Functions on Certain Algebraic Hypersurfaces

It is known that if E is a $$C^{\infty }$$ C ∞ determining set, then E is a Markov set if and only if it has Bernstein’s property. This article provides the equivalent of this result for compact subsets of some algebraic varieties.

NONUNIFORMLY FLAT AFFINE ALGEBRAIC HYPERSURFACES

The relationship between interpolation and separation properties of hypersurfaces in Bargmann–Fock spaces over $\mathbb{C}^{n}$ is not well understood except for $n=1$ . We present four examples of smooth affine algebraic hypersurfaces that are not uniformly flat, and show that exactly two of them are interpolating.

A Module-theoretic Characterization of Algebraic Hypersurfaces

In this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.