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Let $V$ be a vector space. $f_1, \dots, f_{d+1}$ - linearly independent homogenious polynomial of degree $d$ on $V$.

Question Is there a two dimensional subspace $U$ of $V$ such that restriction of my polynomials $f_1|_U , \dots , f_{d+1}|_U$ is still linearly independent (then generates all homogenious polynomials of degree $d$ on $U$)?

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Let $g$ be a homogeneous polynomial of degree $d-1$. $d = dim V$.

$x_1, \dots x_d$ linear functions on $V$.

$f_1 = x_1 g$ , $\dots$, $f_n =x_n g$

Then such polynomials, restricted to $U$ becomes linearly dependent.

Indeed, $U$ satisfies a linear equation $\sum a_i x_i = 0$. Then $\sum a_i f_i|_U=0$

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