最近，Jennifer和Janos在美国数学月刊上发表了一篇文章[1],在其中探讨了多项式在原点和无穷远处消失的速率问题。我这篇文章是对上述文章中关于原点附近消失情况的一个综述，并且包含了我的一些学习体会。 首先，我们明确了一下研究对象，即在原点处有孤立局部极小值零的两个变量的实多项式。 其次，我们研究这样的多项式在原点附近的消失速率。这个速率有一个上界和一个下界。我们发现，上界是容易得到的，它与多项式在原点处的零点阶数保持一致。文章接下来着重对消失速率的下界进行了较为精确的刻画。在计算消失速率下界时，首先引入了一个关键引理。假设引理成立，我们把多项式限制在一个以原点为中心的充分小的正方形上，计算其在这个小正方形上的临界点，然后确定多项式在这个小区域上的最小值，从而得到该多项式在原点附近消失速率的下界。结果表明，这个速率不慢于由多项式次数n决定的n(n-1)阶。 在文章的最后，我们利用预备知识里的西尔维斯特矩阵的知识对关键引理进行了简单的证明。

Recently, Jennifer and Janos published an article [1] in the American Mathematical Monthly, in which they explored the disappearance rate of polynomial at the origin and infinity. This paper is a review of the above article about the case of origin and also contains some of my learning experience. First, we define the object of our study, which is some polynomial in two variables which has isolated local minimum value of zero real at the origin. Second, we study how fast the disappearance rate can be near the origin. We find that the rate of disappearance has an upper bound and a lower bound, and the upper bound is easily proved to be related to the order of the polynomial’s zeros. To estimate the lower bound of the disappearance rate, we first introduce a key lemma. Assuming the key lemma is true, we limit the polynomial to a sufficiently small square which centered at the origin, and then we determine the minimum value of the polynomial in this small area, and obtain the lower bound of the rate of disappearance of the polynomial nearby the origin. It turns out that the rate is slower than the order of n (n-1). Last, we prove the key lemma by using the knowledge of the Sylvester Matrix.