A bad point of a positive semidefinite real polynomial f is a point at which a pole appears in all expressions of f as a sum of squares of rational functions. We show that quartic polynomials in three variables never have bad points. We give examples of positive semidefinite polynomials with a bad point at the origin, that are nevertheless sums of squares of formal power series, answering a question of Brumfiel. We also give an example of a positive semidefinite polynomial in three variables with a complex bad point that is not real, answering a question of Scheiderer.

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关于半正定多项式的坏点

半正定实数多项式f的一个坏点是一个极点出现在f 的所有表达式中作为有理函数的平方和。我们表明三个变量中的四次多项式永远不会有坏点。我们给出了在原点有一个坏点的半正定多项式的例子，它们仍然是形式幂级数的平方和，回答了 Brumfiel 的问题。我们还给出了一个三变量的半正定多项式的例子，它有一个不真实的复数坏点，回答了 Scheiderer 的一个问题。