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In one variable, a polynomial (of any degree) is determined by its values on a finite set of points. More specifically if $p$ is a polynomial of degree $k$, and $x_0 , \dots x_{k}$ are points for which we know the values $\{p(x_{i})\}_{i=0}^{k}$, then we can determine $p$. There are several ways of doing this, but the most elementary is probably by setting

$$Q_i(t):=\prod_{\substack{0\leq j\leq k}\\ \:\:\:j\neq i}\frac{t-x_j}{x_i-x_j}$$ and then it's easy to show that $$p=\sum_{i=0}^{k}p(x_i)Q_i.$$

I want to know if the same is true for polynomials in multiple variables or if there is at least a similar statement. I imagine in multiple variables, polynomials are determined (at the very least) by their values on a full rank lattice in $\mathbb{R}^n$ (e.g., $\mathbb{Z}^n$), though I suppose I don't actually have a proof of that. It just seems highly likely. Theorems, proofs, and references much appreciated.

Rodrigo de Azevedo
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Dan1618
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    Write $$P(x_1,\dots,x_d)=\sum_{i} Q_i(x_1,\dots,x_{d-1}) x_d^i.$$ Then for each given $\mathrm{x}'=(x_1,\dots,x_{d-1})$, we can apply the result for polynomials of single variables to determine the values of $Q_i(\mathrm{x}')$. Now we can recursively apply this idea. Alternatively, if we know that $P(x_1,\dots,x_d)$ has at most $m$ monomial terms, then this will lead us a linear system in $m$ unknowns (corresponding to the coefficients of $P$), and so, we can determine the coefficients. – Sangchul Lee Jul 31 '21 at 04:08
  • Is this the sort of thing (for $2$ variables) you have in mind? From my notes from a combinatorics class from long ago: "Suppose $f(x,y)$ is a polynomial. (1) If $f(x,y)$ vanishes at infinitely many points on a line $L$ in the $xy$-plane, then $f(x,y)$ vanishes at all points on $L$. (2) If $f(x,y)$ vanishes on infinitely many different lines, then it vanishes identically." – bof Jul 31 '21 at 04:09
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    Related $,-,$ Lagrange interpolation of multivariate polynomials and links therein. – dxiv Jul 31 '21 at 04:13
  • @bof How would you prove such a result combinatorially? – Dan1618 Jul 31 '21 at 15:08
  • @SangchulLee Thanks! – Dan1618 Jul 31 '21 at 15:09
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    @Dan1618 Who said anything about proving such a result combinatorially? It was in my combinatorics notes as a Useful Fact From Algebra. It was the reason why some combinatorial identities, proved "combinatorially" for certain integral values of the variables, could be generalized to all real (or complex) values without further ado. – bof Jul 31 '21 at 21:55

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Yes, the multivariate polynomial is uniquely determined by its values on the lattice points.

Reference to a stronger result: Alon, N., Tarsi, M. Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992). https://doi.org/10.1007/BF01204715 , lemma 2.1

(based on an answer of @LinAlgMan to this question Vanishing of a multivariable polynomial on a lattice)

Piotr Śniady
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