An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial
g
(
x
)
∈
C
[
x
]
g(x)\in \mathbb {C}[x]
when its square
g
(
x
)
2
g(x)^2
has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open.
In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations
f
(
x
,
g
(
x
)
)
=
0
f(x,g(x))=0
, where
f
(
x
,
y
)
f(x,y)
is monic of arbitrary degree in
y
y
and has boundedly many terms in
x
x
: we prove that the number of terms of such a
g
(
x
)
g(x)
is necessarily bounded. This includes the previous results as extremely special cases.
We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus
G
m
l
\mathbb {G}_\textrm {m}^l
. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of
G
m
l
\mathbb {G}_\textrm {m}^l
, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials.