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In a paper "G. Kós, L. Rónyai, Alon’s Nullstellensatz for multisets, Combinatorica, 32(5) (2012) 589-605", the authors prove a multiset version of the Cauchy-Davenport Theorem (please see Theorem 14 on page-10 of the paper on arXiv: https://arxiv.org/abs/1008.2901)

In the proof of this theorem, I am not able to understand (in the last paragraph) the meaning of "$f(x, y)$ vanishes at least $r > k + l$ times at $(a, b)$" and also not able to understand how does this conclusion gives a contradiction.

Any help would be appreciated. Thanks in advance!

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  • $\begingroup$ The sentence means that $(a,b)$ is a root of the $f$ with multiplicity at least $r$, with $r>(k+\ell)$. Therefore $f(a,b)=0$, $f'(a,b)=0$, $\cdots$, $f^{(r-1)}(a,b)=0$. Not sure about that conclusion, I haven't read the paper. $\endgroup$
    – Thomas Lesgourgues
    Commented Feb 19, 2023 at 1:00
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    $\begingroup$ That mean the coefficents of all terms of degree less than $r$ in Taylor expansion of $f(x,y)$ at $(a,b)$ are $0$, contradiction because we have nonzero coefficent term $(x-a)^k(x-b)^l$ with $l<r$ $\endgroup$
    – Veronica Phan
    Commented Feb 19, 2023 at 8:23
  • $\begingroup$ @ThomasLesgourgues Thanks! It is understood now. $\endgroup$
    – Rajkumar
    Commented Feb 20, 2023 at 7:37
  • $\begingroup$ @VeronicaPhan Thanks! It is understood now. $\endgroup$
    – Rajkumar
    Commented Feb 20, 2023 at 7:38

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