# When is a polynomial zero mod $p^{e+1}$?

I am reading the following paper:

http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.4233

It comes to the following claim: If $p$ is a prime number, and $Z$ is a multivariate polynomial, then:

$Z$ is zero modulo $p^{e+1}$ iff Z is zero modulo $P$ and $Z \choose p$ is zero modulo $p^e$ where $Z \choose p$ denotes the sum of all possible products of $p$ terms from $Z$ (we avoid using coefficients in writing $Z$ as sum of monomial terms, and repeat terms if necessary)

I fail to see why this is correct.

The "modulo P" is most likely a typo and they mean "modulo p", but P is used in the paper to describes polynomials, so maybe I'm missing something.

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Can you give the page and section numbers for the claim? – Srivatsan Nov 8 '11 at 16:35
Lemma 1, page 5. Thanks. – Gadi A Nov 9 '11 at 8:36