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Let be $p(x_1,...,x_n) \in K[x_1,...,x_n]$ be a polynomial of degree $d$. Suppose there is a $n$-dimensional hyperbox $B = I \times \stackrel{n}{...} \times I = I^n$. Divide $I$ to $d$ segements by $d+1$ points. This creates a lattice of $(d+1)^n$ point on $B$. Suppose that $p$ vanishes on the lattice (that is, for any point $(v_1,...,v_n)$ in the lattice, $p(v_1,...,v_n)=0$). Then we want to show that $p \equiv 0$ is the zero polynomial.

Is this true for $K = \mathbb{R}$? Is it true when $K$ is algebraically closed field (e.g. $K = \mathbb{C}$)? If it is, is there a reference for that proposition?

LinAlgMan
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I think you can do this inductively, as follows. First, look at a 1-dimensional edge $E$ of your box: this contains $d+1$ lattice points, and $p$ is zero at each of them, so $p$ must be zero along the edge. Now look at a 2-dimensional face $F$ containing $E$; it has $d+1$ lines $E_i$ parallel to $E$ (including $E$ itself), and the same argument shows that $p$ must be zero along each of these. But now any line $L$ joining $E \, (=E_1)$ to the opposite edge $E_{d+1}$ intersects all the $E_i$, so $p$ has at least $d+1$ zeros along $L$, hence must be identically zero along $L$. Since this is true for any such $L$, in fact $p$ must be zero along the whole face $F$. Now keep going in the same way.

  • Thank you for the proof. I need to state this result in a paper so I am looking for a reference (book, paper, lecture notes) so I will be able to write "Proposition. (this claim). Proof: see [1]." – LinAlgMan Oct 10 '13 at 09:32
  • OK; I was answering your first question, which says "Is this true?" I don't know a reference, but the proof is short enough to reproduce in a paper. –  Oct 10 '13 at 09:34
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    I found a reference: http://www.tau.ac.il/~nogaa/PDFS/null2.pdf , Lemma 2.1 by Noga Alon. Thank you. – LinAlgMan Oct 16 '13 at 09:40
  • Even a more direct reference based on the suggestion of @LinAlgMan: Alon, N., Tarsi, M. Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992). https://doi.org/10.1007/BF01204715 , lemma 2.1 – Piotr Śniady May 09 '22 at 07:50