Let $P\_n(t) = p\_0 + p\_1 t + \cdots + p\_n t^n$ be a polynomial (with real coefficients) of degree $n$ in the variable $t$. I am interested in the quantity $$\Phi\_n = \min\_{\sum\_{i=1}^n p\_i^2 = 1} \int\_0^1 P\_n^2(t).$$ 
In particular: how fast does $\Phi\_n$ approach zero as $n \rightarrow \infty$? Can anything be said about which polynomials achieve the minimum, or which achieve something close to the minimal $\Phi\_n$?


Note that this can be reformulated as a question about the smallest eigenvalue of a certain matrix. 



Polynomial with the smallest area
This summarizes various comments from above. Write $p=(p\_0,p\_1,\ldots, p\_n)$. Since
$$
\Phi\_n = \min\_{\|p\|\_2 =1} \sum\_{j,k=0}^n \frac{p\_jp\_k}{j+k+1} ,
$$
we can also interpret $\Phi\_n$ as the smallest eigenvalue of the (positive definite) Hilbert matrix $H\_{jk}=1/(j+k+1)$, $j,k=0,1, \ldots , n$.


We find the following asymptotic formula in the literature:
$$
\Phi\_n = K\sqrt{n}\left( 1+\sqrt{2}\right)^{-4n}(1+o(1)), \quad\quad K=\frac{8\cdot 2^{3/4}\pi^{3/2}}{(1+\sqrt{2})^4}
$$
See [here](http://link.springer.com/article/10.1023%2FA%3A1004180718725) and [here.](http://www.jstor.org/stable/2035162?seq=1#page_scan_tab_contents)


