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In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero polynomial that vanishes on $S$ with degree $\leq n|S|^{1/n}$" ($\mathbb F$ is a field).

It seems that if we include the phrase "such that a fraction $r\in[0,1]$ of them satisfy no tangential conditions at those points" then there should be no change in the degree.

However if we include the phrase "such that a fraction $r\in[0,1]$ of them satisfy tangential conditions at those points" then the degree might blow up.

Is there a bound on degree if we also the phrase "such that a fraction $r\in[0,1]$ of them satisfy tangential conditions at those points"? I am intereseted in $r\rightarrow1$.

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  • $\begingroup$ $\mathbb{F}$ is a finite field: am I right? $\endgroup$ – Armando j18eos Sep 22 '18 at 23:09
  • $\begingroup$ @Armandoj18eos No document refers only 'Field' terminology. $\endgroup$ – Freeman. Sep 22 '18 at 23:25

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