Sharp bounds for minimal dependencies of linear-form powers

Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound $$\dim L\le \frac{t+m-2}{m}\mathrm{.}$$ Here ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers ${\mathrm{\ell }}_1^m,\dots ,{\mathrm{\ell }}_t^m$ form a circuit, and $L=\text{Span}\{{\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_t\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\dim L)$; in particular, the affine bound itself is sharp and the optimal leading constant in Bukh’s question is $1\mathrm{/}m$. The proof uses a coding-theoretic translation: the coefficient row space of the powers is the $m$-th Schur power of the coefficient code, and the minimality hypothesis makes this Schur power a full-support hyperplane to which the Schur-product Kneser theorem of Mirandola and Zémor applies. The same method yields flat-concentration and interpolation criteria, a Cayley–Bacharach lower bound, Segre–Veronese and positive-characteristic variants, and Hilbert-function constraints for equality and near-equality in Veronese circuits.