1. INTRODUCTION 3
and sufficient condition of (1.4) is not determined by only faces but also by dual
faces of the Newton polyhedron, which has not appeared explicitly in the graph
case Λ = (e1, · · · , en, Λn+1) or low dimensional case n 2.
Organization. In Chapter 2, we define some useful combinatorial notions. In
Chapter 3, we state the main results. We prove combinatorial lemmas in Chapters
4 and 5. Next, we show preliminary analytic estimates in Chapter 6. The bulk of
the proof of Main Theorems are found in Chapters 7–11.
Notations. For the sake of distinction, we shall use the notations
ı · j = ı1j1 + · · · + ınjn , x, y = x1y1 + · · · + xdyd
for the inner products on
Zn, Rd,
respectively. Note that a constant C may be
different on each line. As usual, the notation A B for two scalar expressions
A, B will mean A CB for some positive constant C independent of A, B and
A B will mean A B and B A .
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