| Abstract: |
| We study the boundary value problem of the heat equation in the space of all real polynomials of two variables $x$ and $t$. Our interest is to determine a polynomial $d(x,t)$ such that for every polynomial $f(x,t)$, there exists a heat polynomial $u(x,t)$ which is equal to $f(x,t)$ on the curve $d(x,t) = 0$ in the $xt$-plane, where heat polynomial means a polynomial which satisfies the heat equation. We give all such $d$ with degree at most two, and prove that there exist no such $d$ of degree greater than 3. The Hermite polynomial $H_n(x)$ plays an essential role in our argument. In particular, the irreducibility of $H_{2n-1}(x)/x$ and $H_{2n}(x)$ in the field of rational numbers is very important. |
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