The threshold degree of a Boolean function
$f\colon\{0,1\}^n\to\{-1,+1\}$ is the least degree of a real
polynomial $p$ such that $f(x)\equiv\mathrm{sgn}\; p(x).$ We
construct two halfspaces on $\{0,1\}^n$ whose intersection has
threshold degree $\Theta(\sqrt n),$ an exponential improvement on
previous lower bounds. This solves an open problem due to Klivans
(2002) and rules out the use of perceptron-based techniques for
PAC learning the intersection of two halfspaces, a central
unresolved challenge in computational learning. We also prove
that the intersection of two majority functions has threshold
degree $\Omega(\log n),$ which is tight and settles a conjecture
of O'Donnell and Servedio (2003).

Our proof consists of two parts. First, we show that
for any nonconstant Boolean functions $f$ and $g,$ the
intersection $f(x)\wedge g(y)$ has threshold degree
$O(d)$ if and only if $\|f-F\|_\infty + \|g-G\|_\infty
< 1$ for some rational functions $F,$ $G$ of degree
$O(d).$ Second, we settle the least degree required for
approximating a halfspace and a majority function to
any given accuracy by rational functions.

Our technique further allows us to make progress on
Aaronson's challenge (2008) and contribute direct sum
theorems for polynomial representations of composed Boolean functions
of the form $F(f_1,...,f_n).$ In particular, we give an improved
lower bound on the approximate degree of the AND-OR tree.

1) Bibliographic information added in Section 3.1 on
complementary upper bounds for approximating composed
Boolean functions, due to Buhrman, Newman, Rohrig, and
de Wolf (2007).

2) Typos corrected in the abstract.

The threshold degree of a Boolean function
$f\colon\{0,1\}\to\{-1,+1\}$ is the least degree of a real
polynomial $p$ such $f(x)\equiv\mathrm{sgn}\; p(x).$ We
construct two halfspaces on $\{0,1\}^n$ whose intersection has
threshold degree $\Theta(\sqrt n),$ an exponential improvement on
previous lower bounds. This solves an open problem due to Klivans
(2002) and rules out the use of perceptron-based techniques for
PAC learning the intersection of two halfspaces, a central
unresolved challenge in computational learning. We also prove
that the intersection of two majority functions has threshold
degree $\Omega(\log n),$ which is tight and settles a conjecture
of O'Donnell and Servedio (2003).

Our proof consists of two parts. First, we show that for any
Boolean functions $f$ and $g,$ the intersection $f(x)\wedge g(y)$
has threshold degree $O(d)$ if and only if $\|f-F\|_\infty +
\|g-G\|_\infty < 1$ for some rational functions $F,$ $G$ of
degree $O(d).$ Second, we settle the least degree
required for approximating a halfspace and a majority function to
any given accuracy by rational functions.

Our technique further allows us to make progress on
Aaronson's challenge (2008) and contribute strong direct product
theorems for polynomial representations of composed Boolean functions
of the form $F(f_1,...,f_n).$ In particular, we give an improved
lower bound on the approximate degree of the AND-OR tree.