Talk:Bernstein's problem

Summary statement is unclear

The article's summary of final results is as follows:

"Combined with the result of Simons, this shows that the analogue of Bernstein's theorem is true in dimensions up to 8, and false in higher dimensions. A specific example is the surface $\{x\in \mathbb {R} ^{8}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}$."

This is unclear because the phrase "up to the integer n" does not specify whether the integer n is included or excluded.

Anyone contributing to mathematics articles in Wikipedia ought to understand English well enough to avoid this kind of ambiguity.

In order to make this clear, the words "up through" should be used. And then just to make it even clearer, the phrase "false in higher dimensions" should be replaced by an explicit description of which dimensions in which it is false.

Furthermore, this particular problem has an additional difficulty: The graph of a real-valued function defined on Rⁿ is a hypersurface in Rⁿ⁺¹. So the article ought to go out of its way to make it ultra-clear what "true in dimension n" means: If it is true in dimension n, does n refer to the dimension of the domain of the function, or the dimension of the Euclidean space of which the graph is a submanifold? I hope someone knowledgeable on this subject will fix this ambiguity in the article.

And please note that the example cited in the above quote from the article is not expressed in terms of the graph of a function.50.203.182.230 (talk) 20:31, 23 October 2019 (UTC)