Let $X$ be an irreducible smooth real algebraic variety of dimension at least $2$ and let $f : U \to \mathbb{R}$ be a function defined on a connected open subset $U \subset X(\mathbb{R})$. Assume that for every irreducible smooth real algebraic curve $C \subset X$, for which $C(\mathbb{R})$ is the boundary of a disc embedded in $U$, the restriction ${f \vert}_{C(\mathbb{R})}$ is continuous and has a rational representation. Then $f$ has a rational representation. This is a significant refinement of a recent result of J. Kollár and the authors. The novelty is that existence of rational representation is tested on a much smaller and more rigid class of curves. We also consider the case where $U$ is not necessarily connected and test rationality on subvarieties of dimension greater than $1$. For semialgebraic functions our results hold under slightly weaker assumptions.

中文翻译：

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实函数的合理表示

假设$ X $是维数至少为$ 2 $的不可约光滑实代数，并且$ f：U \ to \ mathbb {R} $是在连接的开放子集$ U \ subset X（\ mathbb {R }）$。假设对于每个不可约的光滑实数代数曲线$ C \ subset X $，其中$ C（\ mathbb {R}）$是嵌入在$ U $中的光盘的边界，则限制$ {f \ vert} _ { C（\ mathbb {R}）} $是连续的，并具有一个有理表示。则$ f $具有理性表示。这是对J.Kollár和作者的最新结果的重大改进。新颖之处在于，在更小，更严格的曲线类别上测试了理性表示的存在。我们还考虑了$ U $不一定是关联的情况，并检验了维度大于$ 1 $的子变量的合理性。