# What is the name of this kind of polynomial?

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? – Mariano Suárez-Alvarez Sep 21 '10 at 16:46
People call them 0-1 polynomials, I think. – Qiaochu Yuan Sep 21 '10 at 16:58
@Mariano, This is a special case of a monic polynomial, so I thought it might have a specific name. – Neves Sep 21 '10 at 17:10
@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. – Qiaochu Yuan Sep 21 '10 at 17:15
@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. – Mariano Suárez-Alvarez Sep 21 '10 at 22:07

## 1 Answer

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 – Neves Sep 21 '10 at 20:02
Another link The Beauty of Roots, John Baez – Neves Sep 23 '10 at 10:15