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BigM
BigM
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Suppose $f\in C^{\infty}(\mathbb{R}^n)$ and $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are typical counterexamples?

Suppose $f\in C^{\infty}(\mathbb{R}^n)$ and $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are typical counterexamples?

Suppose $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are typical counterexamples?

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BigM
BigM
  • 1.6k
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  • 14
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Suppose $f\in C^{\infty}(\mathbb{R}^n)$ and $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are typical counterexamples?

Suppose $f\in C^{\infty}(\mathbb{R}^n)$ and $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is algebraic. Is $f$ necessarily a polynomial?If not what are typical counterexamples?

Suppose $f\in C^{\infty}(\mathbb{R}^n)$ and $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function. Is $f$ necessarily a polynomial?If not what are typical counterexamples?

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BigM
BigM
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Smooth algebraic functions

Suppose $f\in C^{\infty}(\mathbb{R}^n)$ and $f(x)=\sum_{|\alpha|=0}^{\infty}a_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is algebraic. Is $f$ necessarily a polynomial?If not what are typical counterexamples?