Discrete Comput Geom 8 387-416 (1992) ,. . ^

^ - " ^ Discrete & Coin|Hitatioiml

^» - ^ Discrete & ComiHitatioiml

Geometry

с 1992 Springer Verlag New York ïnc •^

A Cone of Inhomogeneous Second-Order Polynomials"

Robert Erdahl

Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada ErdahlR@QUCDN QueensU CA

Communicated by H S M Coxeter

Abstract . Let ^" be the cone of quadratic functions

Fl / = /о + Z /.^. + Z fij^i^r fij = /л' on Ш" that satisfy the additional condition

F2 /(z)>0, zeZ",

where Z denotes the integers The coefficients and variables are assumed to be real and I <i,j <n The extent to which information on the convex structure of ^" can be used to determine the integer solutions of the equation / = 0 for / g ^" has been studied

The root figure of/e^", denoted Rf, is the set of n-vectors zeZ" satisfying the equation /(z) = 0 The root figures relate to the convex structure of ^" in an obvious way if Ä IS a root figure, then

IS a relatively open face with closure {q e ^"\q(r) = 0, r e K} However, such formulas do not hold for all the relatively open and closed faces, this relates to some subtleties in the structure of ^"

Enumeration of the possible root figures is the central problem in the theory of ^" The group G(Z"), of affine transformations on IR" leaving Z" invanant, is the full symmetry group of ^" Classification of the root figures up to G(Z")-equivalence

* This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Advisory Research Committee of Queen's University