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Let $k, d \in \mathbb{Z}_{\geq 1}$ be positive integers.

I want to prove the following.There exists a large enough $r \in \mathbb{Z}_{\geq 1}$ (depending on $k$ and $d$) and a Zariski open set $U \subseteq \mathbb{R}^{r+1}$ such that the following holds. For any real univariate polynomial $\sigma$ of degree $r$ whose coefficients lie in $U$, and distinct non-constant multivariate polynomials $p_1, \ldots ,p_k \in \mathbb{R}[x_1, \ldots, x_d]$, the multivariate polynomials $\sigma(p_1), \ldots, \sigma(p_k)$ are linearly independent.

Any ideas on how to prove this? Thank you so much in advance.

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    $\begingroup$ Nice question! Note that it suffices to prove the claim for $d = 1$, since if $p_1, \ldots, p_k$ are mutually distinct, then so are $p_1(a_1t, \ldots, a_dt), \ldots, p_k(a_1t, \ldots, a_dt)$ for generic $a_1, \ldots, a_d$. $\endgroup$ Commented Sep 6 at 13:58
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    $\begingroup$ Possibly relevant: D. J. Newman, “Waring’s Problem for the Ring of Polynomials.” It treats the case $\sigma(x)=x^n$ and assumes the $p_i$ are pairwise non-proportional. After applying @pinaki’s suggestion, you may be able to adapt Newman’s Wronskian-based arguments (via the Wronskian matrix). $\endgroup$ Commented Sep 6 at 14:47

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