Codimension of variety versus linear independence of defining homogeneous polynomials

Let $p_1, ..., p_m \colon \mathbb{R}^n \to \mathbb{R}$ be $m < n$ linearly independent homogeneous polynomials each of degree exactly $d$, and assume they have nonzero intersection. Is the codimension of the affine variety determined by $p_1, ..., p_m$ at least $m$?

If this does not hold in general, can we lower bound the codimension of the variety in terms of $d$ (e.g., what if $d=2$)?