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So, in

Size of vector space of polynomial that vanish on some set

the following question is asked. However, I am not convinced by the logic of the answer. Rather than revive an old question, I thought it would be best to ask the question in a slightly more general setting.

Let $X\subset \mathbb{F}_q^n$ and let $M(n,d)$ consist of polynomials of degree at most $d$, such that the degree of any given variable is at most $q-1$. Let $V\subset M(n,d)$ be the subspace consisting of polynomials vanishing on $X$

It is claimed that $dim\ V \geq dim\ M(n,d) - |X|$. Could someone give a hint as to why this is the case?

Aidan
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1 Answers1

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Hint: Let $W$ be the space of functions from $X$ to $\Bbb{F}_q$. Then

  • $\dim W=|X|$.
  • The restriction of polynomial mappings from $\Bbb{F}_q^n$ to $X$ gives a linear transformation $\phi$ from $M(n,d)$ to $W$. What does rank-nullity tell about its kernel?
Jyrki Lahtonen
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