George M. Bergman, The logarithmic limit-set of an algebraic variety, Trans. A.M.S. 157 (1971) 459-469.  MR 43#6209.


The conjecture on p.467 of the paper has been proved :

Robert Bieri and J.R.J.Groves, The geometry of the set of characters induced by valuations, J. reine u. angew. Math. 347 (1984) 168-195.  MR 86c:14001. 

Cf. also

Robert Bieri and J.R.J.Groves, A rigidity principle for the set of all characters induced by valuations, T.A.M.S. 294 (1986) 425-434.  MR 87i:16015. 

The results of the paper itself have been used by group theorists, e.g.,

J. E. Roseblade, Group rings of polycyclic groups J. Pure Appl. Algebra 3 (1973) 307-328.  MR 48#11269. 

Another paper extending the ideas of my paper is

Daniel R. Farkas, The Diophantine nature of some constructions at infinity, Geometry of group representations (Boulder, CO, 1987), 125-129, Contemp. Math., 74.  MR 89m:20008. 

See also the book

Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, v.97, AMS, 2002. viii+152 pp.  MR2003i:13037


The second half of section 3 concerns certain families of subgroups  G  of  Zn  which arise as kernels of homomorphisms to ordered groups.  Clearly, every such subgroup is pure, equivalently (since  Zn  is torsion-free), satisfies  nx ∈ Gx ∈ G  for positive integers  n.  With that case in mind, I unfortunately made some general assertions about families of subgroups of  Zn  that are not true without a purity hypothesis.  In particular, the statement on p.462, end of paragraph slightly above the center, "for  T > S  we have (p1(T), ..., pn(T)) < (p1(S), ..., pn(S))  under lexicographic ordering" is false (e.g., let  n = 1,   S = {6 Z},  T = {2 Z, 3 Z}).  Likewise, in the last sentence of that section, my assertion that  Q  consists of "nontrivial" subgroups, which as defined in that paper must have infinite index in  Zn, does not follow merely from the preceding assertion that  Q < {Zn}. 

There are several ways these problems can be fixed.  The simplest is to define a "subgroup system" to be a finite nonempty family of pure subgroups of  Zn.  Then the above incorrect assertions become correct.  The one modification this entails is in the description of  ST,  which will then consist, not of the sums  G + H  with  G ∈ SH ∈ T,  but of the division-closures of those sums in  Zn

A similar solution is to replace the concept of a "subgroup system" by that of a "subspace system", meaning a finite family of vector subspaces of  Qn.  The general statements about subgroup systems in section 3 become correct for these subspace systems, but a little more work is needed in connecting these results with our hypotheses and conclusions: We must note how to obtain from our homomorphisms on  Zn  subspaces of  Qn,  and then having concluded that that the stabilizer of  I  in  GL(nZ)  stabilizes a nontrivial subspace system  S,  we must note that it therefore stabilizes a nontrivial finite family of subgroups of  Zn,  namely  {G ∩ Zn | G ∈ S}. 

As a third alternative, one could keep the original definition of a "subgroup system" (though it is a finer tool than required), and modify our arguments.  Despite the incorrectness of the proof using lexicographic order, it is true that the lattice of all subgroup systems of  Zn  has ascending chain condition; this is a case of a general lemma proved at the end of this errata-page.  And the fact, needed at the end of section 3, that  Q  consists of subgroups of infinite index in  Zn  can be seen by noting that if it did not, then on dropping all subgroups of finite index, one would have a subgroup system majorizing  Q  under our partial ordering and still compatible with  V

I am indebted to Greg Marks for pointing out the errors corrected above, and a couple of those noted in the following list of minor corrections:

P.461, last line of first full paragraph:  Xα  should be  xα

P.463, top line: After "other elements" add "t", and at the end of the line, change "relation" to "relations". 

P.464, first sentence of proof of Theorem 2: In (2),  &supe  should be  &sube,  and on the next line, after "finally", add "(4)". 

P.465, line after first display: In the first inequality, both occurrences of  max  should be  max(2)  and inversely in the second inequality, both occurrences of  max(2) should be  max. 
  (To see the former inequality, consider two cases: If  |cα xα|  is maximized by the same  α  that maximizes  |xα|,  then all other terms  |cα xα|  involve an  |xα|  term that is  ≤ max(2)(|xα|),  and the result clearly holds.  In the contrary case, the maximum value of  |cα xα|  is itself ≤ M max(2)(|xα|),  hence a fortiori so is the  max(2)  value.  The second inequality is straightforward.)

Throughout the paper, I often wrote "designate" where I would now use "denote". 


The fact that the set of subgroup systems  Zn  has ACC is a special case of the following lemma, probably known to people in the field of partially ordered sets.  I would be grateful to anyone who could give me a reference.  (Note that the antichains of the lemma correspond to the irredundant subgroup systems of the paper.)

Lemma.  Let  P  be any partially ordered set with ACC, and let  FA(P)  be the partially ordered set of finite antichains in  P  (finite subsets with no comparable elements), ordered by making  TS  if every member of  T  is  ≥  some member of  S.  Then  FA(P)  also has ACC. 

Proof.  Let  S0S1 ≤ ...  in  FA(P);  it will clearly suffice to show that  i Si  is finite.  Given  m ≥ 0,  sm  ∈Sm  and  t ∈i Si,  let us write  (m, sm) → t  if for some  n ≥ m  there exists a finite increasing sequence  smsm+1 ≤ ...  ≤ sn = t  with  si ∈Si  for all  i.  By the hypothesis  S0S1 ≤ ...   and the description of our partial ordering on antichains, we see that for every  t ∈i Si  there is some  s0 ∈S0  such that  (0, s0) → t.  So to show  i Si  finite, it will suffice to show that for every  s0 ∈S0,  {t | (0, s0) → t}  is finite.  We shall in fact show that for every  m ≥ 0  and  sm ∈ Sm,  {t | (m, sm) → t}  is finite. 

Assuming the contrary, let us choose  m ≥ 0 and  sm ∈Sm  with  {t | (m, sm) → t}  infinite so as to maximize the element  sm ∈P.  The infinite set  {t | (m, sm) → t}  must contain some element  ≠ sm,  so we may choose  n > m  and  u ∈Sn  distinct from  sm  such that  (m, sm) → u.  In particular  u > sm,  so as  Sn  is an antichain,  Sn  cannot also contain  sm,  hence all elements  sn ∈Sn  that are  ≥ sm  are strictly greater than it.  Hence by the maximality assumption on  sm,  each of those elements  sn ∈Sn  has the property that  {t | (n, sn) → t}  is finite; hence the union of those sets is finite. 

But note that any sequence  smsm+1 ≤ ... ≤ sp  with  si  ∈Si  for all  i  must have the property that all its terms  si  with  i ≥ n  (if any) lie in the abovementioned union; while of course all the terms with  i < n  lie in the finite set  Sm ∪ ... ∪ Sn-1.  Hence all terms of such sequences, i.e., all elements of  {t | (m, sm) → t},  lie in one of these two finite sets, contradicting the assumption that  {t | (m, sm) → t}  is infinite. \qed

Afterthought: If we associate to every finite antichain  S  in  P  the dual ideal (subset of  P  closed under passing to larger elements) that it generates, we get a bijection between the set of finite antichains and the set of finitely generated dual ideals of  P.  The ordering described above then corresponds to the opposite of the inclusion ordering on dual ideals.  Hence, reversing all orderings, the above lemma translates to say that if  P  has DCC, then the set of its finitely generated ideals, partially ordered by inclusion, also has DCC.  It seems most likely to be known in something like this form to people who study ordered sets. 

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