Weighted sums of squares in local rings and their completions, II

Let A be an excellent regular local ring of dimension two, let T be a finitely generated preordering in A, and let [^(T)]{widehat T} be the preordering generated by T in the completion [^(A)]{widehat A} of A. We study the question when the property of being saturated descends from [^(T)]{widehat T} to T, and establish conditions of geometric nature which allow to decide this question. As an application we classify all principal preorderings in A of degree ≤ 3 which are saturated, in the case where A has a real closed residue field. These results have direct implications for nonnegativity certificates for real polynomials on two-dimensional semi-algebraic sets.     

Characterization of completions of reduced local rings

We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring with maximal ideal are (i) or , and (ii) for all , if is an integer of , then .

     

Symbolic squares in regular local rings

In this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal. Our techniques come from residual intersections, integral closures of ideals, and differentials. Received January 14, 1997; in final form March 20, 1997     

Quasi-direct sums of rings and their radicals

Let A, B be rings and P a radical property. Call B an A-Algebra if B is an A-bimodule such that (ba)b₁ = b(ab₁), (bb₁)a = b(b₁a), a(bb₁) = (ab)b₁ for any a ∈ A and any b,b₁ ∈ B. A ring R, written as R = A ? B, is called a quasi-direct sum of (A, B) if A is a subring of R, B is an ideal of R and R is a direct sum of A and B as additive groups. The following results are obtained: 1. A quasi-direct sum of (A, B) is uniquely determined by an A-Algebra B (up to isomorphism); 2. The P-radical of the Algebra B is the same as the P-radical of the ring B; 3. P(A ? B) = P(A) +(B) if and only if P(A)B + BP(A) ? P(B); 4. If B has an identity e then P(A ? B) = P(A)(1?e) + P(B); 5. If P(Z) = 0 for the integer ring Z, then P(M_n(R)) = M_n(P(R)) holds for all rings R if and only if the above equality holds for all unitary rings R. In addition, some relationships of radicals between rings (or algebras over a field, semigroup algebras, etc.) and their corresponding identity extensions are discussed.     

Zero-divisors in completions of non-commutative rings

We show that it is possible for a regular element of a noncommutative Noetherian ringR to become a zero-divisor in theM-adic completion ofR for a maximal idealM ofR.     

I-Adic completions of non-commutative rings

Some sufficient conditions are given for theI-adic completion of a non-commutative ringR to be Noetherian. The case considered is whenI is polycentral or has a normalising set of generators. In the polycentral case we use an associated graded ring argument. This paper was written while the author was a Fellow of the Institute for Advanced Studies. The Hebrew University, Mount Scopus, Jerusalem, Israel.     

On completions of non-commutative noetherian rings

It is shown that if I is an ideal of a ring R ,and I has a centralising set of generators then the I-adic completion [Rcirc] is left noetherian if either R/I is left artinian or R is left noetherian.     

On sums of squares

We show that the fourth order form in five variables,

$\text{∑}\text{i=1}\text{5}\text{∏}\text{i=≠i}\text{(x}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{i}}\text{)}$

, is nonnegative, but cannot be written as a sum of squares of quadratic forms.     

Nonnegative functions as squares or sums of squares

We prove that, for n?4, there are C^∞ nonnegative functions f of n variables (and even flat ones for n?5) which are not a finite sum of squares of C² functions. For n=1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f=g². We prove that, in general, one cannot require a better regularity than g∈C¹. Assuming that f vanishes at all its local minima, we prove that it is possible to get g∈C² but that one cannot require any additional regularity.     

Greedy sums of distinct squares

When a positive integer is expressed as a sum of squares, with each successive summand as large as possible, the summands decrease rapidly in size until the very end, where one may find two 's, or several 's. We find that the set of integers for which the summands are distinct does not have a natural density but that the counting function oscillates in a predictable way.

     

Rectangles as sums of squares

In this paper we examine generalisations of the following problem posed by Laczkovich: Given an n×m rectangle with n and m integers, it can be written as a disjoint union of squares; what is the smallest number of squares that can be used? He also asked the corresponding higher dimensional analogue. For the two dimensional case Kenyon proved a tight logarithmic bound but left open the higher dimensional case. Using completely different methods we prove good upper and lower bounds for this case as well as some other variants.     

Matrices as sums of squares

How many squares are needed to represent elements in a matrix ring? A matrix over a field of characteristic two is a sum of two squares if and only if its trace is a square, otherwise it is not a sum of squares. Any proper matrix over a field of characteristic not two is always a sum of three squares. If the order of a matrix is even the matrix is a sum of two squares, but an odd order matrix which is q times the identity matrix is a sum of two squares if and only ifq is a sum of two squares in the field. Matrices of order 2,3 and 4 over the integers can always be written as the sum of three squares.     

Nil properties for rings which are sums of their additive subgroups

Suppose that a ring R is a direct sum of a finite number of its additive subgroups, and the union of these subgroups is closed under multiplication. We show that if all rings among these subgroups are nilpotent (left T-nilpotent, locally nilpotent or Baer radical), then the whole ring R satisfies the same property.     

Sparsity in sums of squares of polynomials

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy.A considerable part of this work was conducted while this author was visiting Tokyo Institute of Technology. Research supported by Kosef R004-000-2001-00200Mathematics Subject Classification (1991): 90C22, 90C26, 90C30