# Polynomials passing through points with tangential conditions

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 $$r\rightarrow1$$.

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