Let S=k[x~ ..... x,] be a polynomial ring over an infinite field k, and let 1 be a homogeneous ideal of S.An algorithm for computing the (first) syzygies of I is due independently to Spear [Spe77] and Schreyer [Sch80]: One chooses an ordering on the monomials of S, and then constructs a monomial ideal in(1) generated by the lead terms of all elements of I, in(l) can be viewed as the limit of 1 under the action of a 1-parameter subgroup of GL(n) on the Hilbert scheme [Bay 82], so in(1) occurs as the special fiber of a flat family whose general fiber is isomorphic to I. It follows from a well-known criterion for flatness [Art 76] that each syzygy of in(l) can be lifted to a syzygy of I; the set of syzygies thus obtained can be trimmed to give a complete set of minimal syzygies of I.The rnonomial ideal in(I) was first studied The following problem arises in using this syzygy algorithm in practice: in(l) can have minimal generators and syzygies in degrees higher than any minimal generator or syzygy of I. In this situation, computations in these higher degrees are unnecessary; one should compute the generators and syzygies of in(1) in only those degrees necessary to find all minimal syzygies of 1.In order to modify the syzygy algorithm to take advantage of this observation, one would like a criterion for determining when all minimal syzygies of I have been found. This problem appears to be intractable at present. However, the question of bounding the degrees of the minimal jth syzygies of I, for all j, is tractable. Recall that I is defined to be m-regular if the jth syzygy module of I is generated in degrees 0 ([Mum66], [EiGo 84]).The regularity of I, reg(l), is defined to be the least m for which I is m-regular. We have reg(in (1) We give in w a criterion for 1 to be m-regular, which depends only on computations in the finite vector spaces S,, and S"+I of polynomials of degrees m, m+l:If one can find h 1 .... ,hieS 1, so that the subspaces ((I, h 1 ..... hi-1) : hi)m and (I, hi, ..., h i 1),. are equal for 1 s j, and (I, h 1 .... , hj)m=Sm, then I is m-regular. Furthermore, if I is m-regular, then a generic choice of h~, ..., hje S 1 will satisfy these conditions.One could use this result to terminate syzygy computations early, in cases where reg(in(I))>reg(I). However, further study reveals a close connection between this result and a particular order on the monomials of S, the reverse lexicographic order. This order is used to compute saturations in [Bay 82]. The reverse lexicographic order was then studied in characteristic zero, in generic coordinates, by several authors. It is observed in [Laz83] that under this hypothesis in low dimensions, the generators of in(I) are of particularly low degree. In [Giu84], this hypothesis is further studied, and a worst-case upper bound on the degrees of generators of in(l) is obtained, which improves Hermann's corresponding bound for ideal membership [Her 26]. In [Ang84], independent of a preliminary version of our results, it is shown that under this hypoth...

Abstract.This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called ^-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of ^-trajectories and /1-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and ^-trajectories. It shows that projective transformations map ^-trajectories into f-trajectories. It presents Karmarkar's interpretation of /1-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 = ^2"_x dx¡dxj/xf restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of /1-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm.These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that /"-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesies (actually distinguished chords) of a geometry isometric to a Hubert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The /1-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesies of a geometry isometric to Euclidean geometry.

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