A Note on the Dubois-Efroymson Dimension Theorem

1989 ◽  
Vol 32 (1) ◽  
pp. 24-29 ◽  
Author(s):  
Wojciech Kucharz
Keyword(s):  
Algebraic Subset ◽  
Algebraic Set ◽  
Regular Functions ◽  
The Ideal

AbstractLet X ⊂ Rnn be an irreducible nonsingular algebraic set and let Z be an algebraic subset of X with dim Z ≦ dim X — 2. In this paper it is shown that there exists an irreducible algebraic subset Y of X satisfying the following conditions: dim Y = dim X — 1, Z ⊂ Y and that the ideal of regular functions on X vanishing on Y is principal.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

scholarly journals A remark on projective modules

1987 ◽  
Vol 36 (3) ◽  
pp. 515-520
Author(s):  
Wojciech Kucharz
Keyword(s):  
Projective Modules ◽  
Real Numbers ◽  
Algebraic Subset ◽  
Regular Functions ◽  
The Ideal

Let R denote the field of real numbers and let A be the ring of regular functions on Rn, that is the localization of R[T1 …, Tn] with respect to the set of all polynomials vanishing nowhere on Rn. Let X be an algebraic subset of Rn and let I(X) be the ideal of A of all functions vanishing on X. Assume that X is compact and nonsingular and k = codim X = 1, 2, 4 or 8. we prove here that if the A/I(X)-module I(X)/I(X)2 can be generated by k elements, then there exist a projective A-module P of rank k and a homomorphism from P onto I(X).

scholarly journals On the Cardinality of A Semi-Algebraic Set

1994 ◽  
Vol 1 (3) ◽  
pp. 277-286
Author(s):  
G. Khimshiashvili
Keyword(s):  
Quadratic Forms ◽  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraic Set

Abstract It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.

scholarly journals Normalité de certains anneaux déterminantiels quantiques

1999 ◽  
Vol 42 (3) ◽  
pp. 621-640 ◽  
Author(s):  
Laurent Rigal
Keyword(s):  
Residue Class ◽  
Quantum Algebras ◽  
Regular Functions ◽  
Maximal Orders ◽  
Ring Of Fractions ◽  
Prime Factors ◽  
The Matrix ◽  
Residue Class Ring ◽  
Class Ring ◽  
The Ideal

Let Kq[X] = Oq(M(m, n)) be the quantization of the ring of regular functions on m × n matrices and Iq (X) be the ideal generated by the 2 × 2 quantum minors of the matrix X=(Xij)l≤i≤m, I≤j≤n of generators of Kq[X]. The residue class ring Rq(X) = Kq[X]/Iq(X) (a quantum analogue of determinantal rings) is shown to be an integral domain and a maximal order in its divisionring of fractions. For the proof we use a general lemma concerning maximalorders that we first establish. This lemma actually applies widely to prime factors of quantum algebras. We also prove that, if the parameter isnot a root of unity, all the prime factors of the uniparameter quantum space are maximal orders in their division ring of fractions.

HERMITIAN COMPLEXITY OF REAL POLYNOMIAL IDEALS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250026 ◽  
Author(s):  
JOHN P. D'ANGELO ◽  
MIHAI PUTINAR
Keyword(s):  
Necessary Conditions ◽  
Complex Geometry ◽  
Polynomial Optimization ◽  
Symmetric Polynomial ◽  
Holomorphic Maps ◽  
Real Polynomial ◽  
Algebraic Subset ◽  
Algebraic Set ◽  
Holomorphic Embedding

We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of Cn. This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls.

scholarly journals On the structure of local cohomology modules for monomial curves in

1994 ◽  
Vol 136 ◽  
pp. 81-114 ◽  
Author(s):  
H. Bresinsky ◽  
F. Curtis ◽  
M. Fiorentini ◽  
L. T. Hoa
Keyword(s):  
Local Cohomology ◽  
Closed Field ◽  
Ideal Sheaf ◽  
Homogeneous Ideal ◽  
Local Cohomology Module ◽  
Algebraic Set ◽  
Monomial Curves ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Our setting for this paper is projective 3-space over an algebraically closed field K. By a curve C ⊂ is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Let be the Hartshorne-Rao module of finite length (cf. [R]). Here Z is the set of integers and ℐc the ideal sheaf of C. In [GMV] it is shown that , where is the homogeneous ideal of C, is the first local cohomology module of the R-module M with respect to . Thus there exists a smallest nonnegative integer k ∊ N such that , (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown that k = 0 if and only if C is arithmetically Cohen-Macaulay and C is arithmetically Buchsbaum if and only if k ≤ 1. We therefore have the following natural definition.

K-theory of real algebraic surfaces and threefolds

1989 ◽  
Vol 106 (3) ◽  
pp. 471-480 ◽  
Author(s):  
J. Bochnak ◽  
W. Kucharz
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Algebraic Surfaces ◽  
Real Algebraic Variety ◽  
Projective Modules ◽  
Algebraic Subset ◽  
Witt Group ◽  
Regular Functions ◽  
Real Algebraic Surfaces ◽  
Continuous Complex

LetXbe an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Letdenote the ring of regular functions onX([6], chapter 3). (IfXis an algebraic subset of ℝnthenis comprised of all functions of the formf/g, whereg, f: X→ ℝ are polynomial functions withg−1(O) = Ø.) In this paper, assuming thatXis compact, non-singular, and that dimX≤ 3, we compute the Grothendieck groupof projective modules over(cf. Section 1), and the Grothendieck groupand the Witt groupof symplectic spaces over(cf. Section 2), in terms of the algebraic cohomology groupsandgenerated by the cohomology classes associated with the algebraic subvarieties ofX. We also relate the groupto the Grothendieck groupKO(X) of continuous real vector bundles overX, and the groupsandto the Grothendieck groupK(X)of continuous complex vector bundles overX.

Localization of Intracellular Antigens in thin Sections with Ferritin Labeled Antibody

1970 ◽  
Vol 28 ◽  
pp. 174-175
Author(s):  
M.S. Shahrabadi ◽  
T. Yamamoto
Keyword(s):  
Bovine Serum Albumin ◽  
Bovine Serum ◽  
Antigenic Determinants ◽  
Conjugate Prior ◽  
Thin Sections ◽  
Specific Attachment ◽  
Intracellular Antigen ◽  
Antigen Antibody ◽  
Ideal Method ◽  
The Ideal

The technique of labeling of macromolecules with ferritin conjugated antibody has been successfully used for extracellular antigen by means of staining the specimen with conjugate prior to fixation and embedding. However, the ideal method to determine the location of intracellular antigen would be to do the antigen-antibody reaction in thin sections. This technique contains inherent problems such as the destruction of antigenic determinants during fixation or embedding and the non-specific attachment of conjugate to the embedding media. Certain embedding media such as polyampholytes (2) or cross-linked bovine serum albumin (3) have been introduced to overcome some of these problems.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

X-Ray Analysis of Frozen Hydrated Sections:The Unconventional Approach

1982 ◽  
Vol 40 ◽  
pp. 754-757
Author(s):  
R. Beeuwkes ◽  
A. Saubermann ◽  
P. Echlin ◽  
S. Churchill
Keyword(s):  
Ratio Method ◽  
Frozen Sections ◽  
Technical Problems ◽  
Sample Handling ◽  
X Ray ◽  
Common Group ◽  
Ideal Method ◽  
Classic Paper ◽  
Physical Principles ◽  
The Ideal

Fifteen years ago, Hall described clearly the advantages of the thin section approach to biological x-ray microanalysis, and described clearly the ratio method for quantitive analysis in such preparations. In this now classic paper, he also made it clear that the ideal method of sample preparation would involve only freezing and sectioning at low temperature. Subsequently, Hall and his coworkers, as well as others, have applied themselves to the task of direct x-ray microanalysis of frozen sections. To achieve this goal, different methodological approachs have been developed as different groups sought solutions to a common group of technical problems. This report describes some of these problems and indicates the specific approaches and procedures developed by our group in order to overcome them. We acknowledge that the techniques evolved by our group are quite different from earlier approaches to cryomicrotomy and sample handling, hence the title of our paper. However, such departures from tradition have been based upon our attempt to apply basic physical principles to the processes involved. We feel we have demonstrated that such a break with tradition has valuable consequences.

Sign in / Sign up

Export Citation Format

Share Document