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I'd like to know the name of this kind of polynomial

$p(x)=x^n+a_{1}x^{n-1}+\ldots+a_{n-1}x+1$

where the $a_{i}\in\lbrace0,1\rbrace$.

Thanks.

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    Why would they have a special name?2010-09-21
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    People call them 0-1 polynomials, I think.2010-09-21
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    @Mariano, This is a special case of a monic polynomial, so I thought it might have a specific name.2010-09-21
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    @A.Neves: but there are plenty of good reasons to privilege monic polynomials, e.g. their definition is invariant under translation, they are closely related to integral extensions, etc. This definition does not really have the same kind of conceptual properties.2010-09-21
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    @Qiaochu: 0,1 polynomials don't include the p(0)=1 constraint. For that the name *Newman polynomial* is frequently used - e.g. see the references in my post.2010-09-21
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    @A.Neves, sure. But there are other special cases of monic polynomials (say, those whose coefficients grow from $1$ to some integer $k$ and then back to $1$...). Now, it turns out that they do have a special name, but I wanted to know why did *you* expect them to have a name :)2010-09-21
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    @Mariano, I was playing with cyclotomic equations and I wondered where the roots would go if I deleted some of the terms.2010-09-21
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    @A.Neves, if that is the case, you may be interested in knowing (or, of course, you may already know!) that cyclotomic polynomials do have coefficients which are not 0 or 1 (or -1). There is a 2 in the 105th cyclotomic polynomial.2010-09-21

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They're known as Newman polynomials. They are often studied in contexts where one is interested in learning what interesting consequences result from placing such restrictions on the coefficients, for example see here. Erdos and Littlewoood posed several questions about the effects this has on the minimum modulus of the polynomial on the unit circle, e.g. see this paper.

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    Here is an interesting paper ["Zeros of polynomials with 0,1 coefficients", A. M. Odlyzko and B. Poonen](http://retro.seals.ch/digbib/view;jsessionid=508577B1E580B9C6A779A3ECA50D6EF6?rid=ensmat-001:1993:39::181)2010-09-21
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    Another link [The Beauty of Roots, John Baez](http://math.ucr.edu/home/baez/roots/)2010-09-23