It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general ($C^{\infty}$) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as $u(x,y)x^a y^b +$ flat function, where $u(0,0)\neq 0$ and $a,b$ are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each cases. Our results show new interesting phenomena in one of these cases. Actually, when $a -1/a$ and their poles on the half-plane are contained in the set $\{-k/b:k\in\mathbb{N}$ with $k

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光滑模型情况下局部 zeta 函数的亚形

众所周知，与实解析函数相关的局部 zeta 函数可以作为亚纯函数解析地延续到整个复平面。但是，在一般 ($C^{\infty}$) 光滑函数的情况下，亚纯扩展问题并不明显。事实上，最近已经表明存在特定的光滑函数，其局部 zeta 函数具有不同于极点的奇点。为了理解光滑情况下亚纯扩展的情况，我们研究一个简单但本质上很重要的情况，其中各自的函数表示为 $u(x,y)x^ay^b +$ flat 函数，其中$u(0,0)\neq 0$ 和 $a,b$ 是非负整数。在将平面函数分为四种类型后，我们精确地研究了每种情况下局部 zeta 函数的亚纯扩展。我们的结果在其中一种情况下显示了新的有趣现象。实际上，当 $a -1/a$ 和它们在半平面上的极点包含在集合 $\{-k/b:k\in\mathbb{N}$ 中时， $k