论文标题
hartogs-type定理中的实际代数几何形状，我
Hartogs-type theorems in real algebraic geometry, I
论文作者
论文摘要
令f：x-> r为在r^n中连接的非源性真实代数集x上定义的函数。 We prove that regularity of f can be detected on either algebraic curves or surfaces in X. If dimX>1 and k is a positive integer, then f is a regular function whenever the restriction f|C is a regular function for every algebraic curve C in X that is a C^k submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity.此外，在后一种情况下，C的奇异性等于平面曲线的奇异性，由等式x^p = y^q定义，对于某些素数p <q。如果DIMX> 2，则每当限制f | s是X中每个非主化代数表面s的常规函数时，F是常规函数，对单位2-Sphere均具有同型X中的X表面s。我们还为x不一定连接的这些结果提供了合适的版本。
Let f:X-->R be a function defined on a connected nonsingular real algebraic set X in R^n. We prove that regularity of f can be detected on either algebraic curves or surfaces in X. If dimX>1 and k is a positive integer, then f is a regular function whenever the restriction f|C is a regular function for every algebraic curve C in X that is a C^k submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x^p=y^q for some primes p<q. If dimX>2, then f is a regular function whenever the restriction f|S is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.
