Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following equivalent hold for a polynomial $p \in \mathbb R_d[t]$?

$$p \in S \iff \int_0^1 p(t) q(t) \ge 0 \; \forall q \in S.$$


Here is a counter-example for $d=1$. In this simple case $S$ is just the set of linear combinations of $x$ and $1-x$ with nonnegative coefficients.


$$\int_0^1 p(x)x\ dx=\frac12$$

$$\int_0^1p(x)(1-x)\ dx=0$$

Then $\int_0^1pq\ge0\ \forall q\in S$, but $p\notin S$.

| cite | improve this answer | |
  • $\begingroup$ Thanks! does this change if we require $q$ to be of degree 2d (or some other upperbound)? $\endgroup$ – maroxe Apr 2 '18 at 6:39
  • $\begingroup$ With $q=(x-1)^2$ and $p=3x-1$, I find $\int pq=-1/12$. $\endgroup$ – Jean Duchon Apr 3 '18 at 9:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.