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Are there some interesting properties of polynomials with integer coefficients of degree $2^n$ which satisfy $\mid P(x) \mid \le \frac{1}{2^k}$ ?

I know that their coefficients are bounded and the bound is $4e^d$ where $d$ is the degree. As the coefficients are bounded, the roots are also bounded.

The extremal version of the problem is "Integer Chebyshev" problem.

I also have some more properties like $P(0) = 0$, $P(1) = 0$, $P(-1) = 0$, and any $k$ roots of the polynomial can be known, where $k$ is bounded by polynomial of $n$, but the degree is $2^n$.

Are there more properties about distribution of their zeroes, how their derivatives behave, how these polynomials behave outside $[-1,1]$ ?

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What is the relation between $n$ and $k$? –  Gerry Myerson Mar 10 '13 at 9:51
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Looks like this would be a pretty interesting question if one could figure out exactly what it's asking. It'something about the kinds of polynomials mentioned in the title, but, after that, I'm lost. –  bubba Mar 10 '13 at 10:19
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Is it important that the degree is a power of 2 ?? What is the significance of the inequality on the second line? Can't we just consider polynomials that have a small (or minimal) sup-norm? –  bubba Mar 10 '13 at 10:22
    
The Chebyshev polynomials $T_n(x)$ all have integer coefficients. I'm guessing that the polynomials you're looking for have to be multiples of $T_n(x)$. It appears to me that $kT_n(x)$ will always have non-integer coefficients if $k<1$. I can't prove this when $n$ is odd, but it's obvious when $n$ is even because the constant term in $T_n(x)$ is then 1. –  bubba Mar 10 '13 at 13:46
    
to simplify the problem:-what are the interesting properties of polynomials with integer coefficients taking absolute value <1 in the entire interval -1 to 1. chebychev polynomials comes closest to zero, but these polynomials are close to zero, but they are not necessarily chebyshev i guess. i think we can characterize them using knowledge of analytic theory of polynomials ,approximation theory and algebra –  Amit Sinhababu Mar 10 '13 at 14:08
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1 Answer

It is too long for a comment so I will post it as an answer. After some discussion in the comments the question had been reformulated as follows:

What are the "interesting" properties of the polynomial with integer coefficients such that $\|P_n\|_{C[-1,1]}\le 1.$

Since Chebyshev polynomial satisfies this condition and is an extremal polynomial for many max/min problems in approximation theory it is quite hard to summarize everything. For a good account of such problems see http://books.google.ca/books/about/Analytic_Theory_of_Polynomials.html?id=FzFEEVO3PXYC&redir_esc=y

and http://books.google.ca/books?id=386CC7JnuuwC&printsec=frontcover&dq=Borwein+Erdelyi&hl=en&sa=X&ei=ez2LUbLKCI34rAHS6ID4CA&ved=0CDIQ6AEwAA

Since you mentioned behaviour of derivative, I will give one example here:

Markov's inequality: $\|P_n\|_{C[-1,1]}\le 1$ implies $\|P_n'\|_{C[-1,1]}\le n^2.$ One can generalize this result to higher derivatives.

bubba's observation is also correct: if $0<k<1$ then at least one of the coefficients of $kT_n(x)$ is non integer. Indeed, if this is not the case, $k\cdot 2^n$ has to be integer and therefore $k=\frac{p}{2^m}$ $m\ge 1.$ But using recurrence relation $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ and induction it is easy to show that at least of the coefficients of $T_n$ is odd. This provides a contradiction.

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