Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Step roots of Littlewood polynomials and the extrema of functions in the
  Takagi class   We give a new approach to characterizing and computing the set of global
maximizers and minimizers of the functions in the Takagi class and, in
particular, of the Takagi--Landsberg functions. The latter form a family of
fractal functions $f_\alpha:[0,1]\to\mathbb R$ parameterized by
$\alpha\in(-2,2)$. We show that $f_\alpha$ has a unique maximizer in $[0,1/2]$
if and only if there does not exist a Littlewood polynomial that has $\alpha$
as a certain type of root, called step root. Our general results lead to
explicit and closed-form expressions for the maxima of the Takagi--Landsberg
functions with $\alpha\in(-2,1/2]\cup(1,2)$. For $(1/2,1]$, we show that the
step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root,
then the set of maximizers of $f_\alpha$ is an explicitly given perfect set
with Hausdorff dimension $1/(n+1)$, where $n$ is the degree of the minimal
Littlewood polynomial that has $\alpha$ as its step root. In the same way, we
determine explicitly the minima of all Takagi--Landsberg functions. As a
corollary, we show that the closure of the set of all real roots of all
Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$.

23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ