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Summary: DIGIT PATTERNS AND THE FORMAL ADDITIVE GROUP
GREG W. ANDERSON
Abstract. We prove an elementary result concerning the relationship be
tween the multiplicative groups of the coordinate and endomorphism rings of
the formal additive group over a field of characteristic p > 0. Both statement
and proof of the main result involve the combinatorics of base p representations
of positive integers in a striking way.
1. Introduction
Our main result (Theorem 1.2 below) concerns the relationship between the mul
tiplicative groups of the coordinate and endomorphism rings of the formal additive
group over a field of characteristic p > 0. Our result is elementary and does not
require a great deal of apparatus for its statement. Both statement and proof of the
main result involve the combinatorics of base p representations of positive integers
in a striking way.
1.1. Statement of the main result.
1.1.1. Rings and groups of power series. Fix a prime number p and a field K of
characteristic p. Let q be a power of p. Consider: the (commutative) power series
ring
K[[X ]] = ( #
X i=0
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