No abstract

This note improves, in two respects, the results of §3.6 of my paper The hyper surface cross ratio. 1 There it is shown that the number c n of independent hypersurface cross ratios that can be formed of 2n forms in n variables is 2 for n~ 2, 5 for w = 3, and 14 for n = 4. The proof employs the relations between cross ratios obtained by some simple permutations of the forms; let R be the set of these relations. It is remarked that the cross ratios of 2n -1 forms in n variables, and of In -1 forms in n -1 variables, are connected by the same relations as the cross ratios of In forms in n variables, as far as these are consequences of the relations R, a "perhaps void restriction." We now prove that c n s =C2n,n/(n+l) 1 and that the restriction is in fact void, so that a complete knowledge of the relations between the cross ratios of 2n-l forms, of 2n forms, and of 2w+l forms in n variables Is obtained.2 The corresponding theorems for generalized intersections and one more variable are established at the same time.The same facts hold for a general class of function ratios, which includes hypersurface cross ratios and generalized intersections as very special cases. The number c n of independent function ratios has a simple combinatorial meaning, and appears also as the number of partitions of a polygon by non-intersecting diagonals into triangles, or of a cyclically arranged set into non-interlaced subsets, as the number of possibilities of never losing majority (in an election or a game 8 ), and as the number of different products of given terms in a given order, in a non-associative multiplication. For the combinatorial formula, seven proofs are given, six extended to generalizations. 352

In this note a constructive criterion for the existence of a Euclidean algorithm within a given integral domain is derived, and from among the different possible Euclidean algorithms in an integral domain one is singled out. The same is done for "transfinite" Euclidean algorithms. The criterion obtained is applied to some special rings, in particular rings of quadratic integers. By an example it is shown that there exist principal ideal rings with no Euclidean algorithm. Finally, different sets of axioms for the Euclidean algorithm and related notions are compared, and the possible implications for the classification of principal ideal rings, and other integral domains, indicated.The question of the relationship between different Euclidean algorithms in the same integral domain was raised (orally) by O. Zariski.1. The derived sets. Let Q be an integral domain. A subset P of Q -0 (Q except zero) shall be called a product ideal if P(Q -0)QP.For any subset S of Q, the set B of all b in Q for which there exists an a in Q such that a+bQQS is called the total derived set of S, and the intersection Bl^iS is called the derived set S'. With S also S' is a product ideal. If SiQS, then S{ QS'.A Euclidean algorithm (or process) is given by a norm |a| defined in Q -0, with positive integral (or zero) values and such that \a\ è|&| for b dividing a and that for any b in Q -0 and any a not divisible by b there exist q and r in Q satisfying a = qb+r t \r\ <\b\.Let P it i = 0, 1, 2, • • • , be the set of all b in Q with \b\ ^i. Obviously Pi is a product ideal. For any b in P{, let a be an element with a+bQQPi, whence a -bq^O and (for any r~a -bq with |H |&|è*+l; we see that P!QP i+1 . Conversely, given a sequence Q -0 = Po2-Pi2 • • • of product ideals with empty intersection CiPi such that Pi QPi+i, the norm defined by \b\ -i for every b in Pi -P i+ i will fulfil the conditions for a Euclidean algorithm. Hence there is a one-one correspondence between sequences of this kind and Euclidean algorithms.If for another Euclidean algorithm, with the sequence P t -, always PiQ7i, we say that the first algorithm is the faster one (under cer-

0.1. Not later than 1933 I made the following conjecture, originally in the form of a statement on the minors of a matrix(1). T<¡. Any n points in d-space that are not on one hyper plane determine at least n connecting hyperplanes.Ti is trivial. It is easy to see (4.3) that Td is a consequence of Td_i and Ud-Any n points in d-space that are not on one hyper plane determine at least one ordinary hyperplane, that is, a connecting hyperplane on which all but one of the given points are on one linear (d -2)-space.In particular, T2: n not collinear points are connected by at least n straight lines, is true if U2 holds: for n not collinear points there is a straight line through only two of them. Now the nine inflexions of a plane cubic show that U2 does not hold in the complex plane. Nevertheless H. Hanani gave in 1938 a combinatorial proof of T2 for every (not only the real) projective plane. A greatly simplified version of this proof is given in 4.4. In 1939 A. Robinson proved U2 for the real plane; in 1943 I found another very short proof (respectively the second and first proof in 1

0.1. Not later than 1933 I made the following conjecture, originally in the form of a statement on the minors of a matrix(1). T<¡. Any n points in d-space that are not on one hyper plane determine at least n connecting hyperplanes.Ti is trivial. It is easy to see (4.3) that Td is a consequence of Td_i and Ud-Any n points in d-space that are not on one hyper plane determine at least one ordinary hyperplane, that is, a connecting hyperplane on which all but one of the given points are on one linear (d -2)-space.In particular, T2: n not collinear points are connected by at least n straight lines, is true if U2 holds: for n not collinear points there is a straight line through only two of them. Now the nine inflexions of a plane cubic show that U2 does not hold in the complex plane. Nevertheless H. Hanani gave in 1938 a combinatorial proof of T2 for every (not only the real) projective plane. A greatly simplified version of this proof is given in 4.4. In 1939 A. Robinson proved U2 for the real plane; in 1943 I found another very short proof (respectively the second and first proof in 1