Let me start with two observations.

- In the classification of quadratic forms with rational coefficients, one has the following statement: a quadratic form in five indeterminates represents $0$ over $\mathbb Q$ if and only if it represents $0$ over $\mathbb R$.
- For every $n\ge2$, there exists a division ring $R_n$ of dimension $n^2$ over its center $\mathbb Q$. There is a 'norm' over $R_n$, which is an $n$-form (homogeneous polynomial of degree $n$) with rational coefficients, multiplicative over $R_n$. It is an example of $n$-form in $n^2$ indeterminates that represents $0$ over $\mathbb R$ but does not over $\mathbb Q$.

Question. Is it true that for every $n\ge2$, an $n$-form with rational coefficients in $n^2+1$ indeterminates represents $0$ over $\mathbb Q$ if and only if it represents $0$ over $\mathbb R$ ?

A positive answer would say that the norm in a division ring is a maximal (in terms of the number of indeterminates) example of an $n$-form representing zero over $\mathbb R$ but not over $\mathbb Q$.

*Edit*. After Laurent's answer, I see that my question was close to Artin's conjecture, that Terjanian's example disproves. However, for fixed degree $n$, Artin's conjecture is true for almost every prime $p$: every $n$-form in $n^2+1$ indeterminates represents zero over $Q_p$.