A Product Formula for Minimal Polynomials and Degree Bounds for Inverses of Polynomial Automorphisms

ABSTRACT. By means of Galois theory, we give a product formula for the minimal polynomial G of {fo, fi , ... ,fn) c K[xl , ... ,xn] which contains n algebraically independent elements, where K is a field of characteristic zero. As an application of the product formula, we give a simple proof of Gabber's degree bound inequality for the inverse of a polynomial automorphism.published_or_final_versio

Automorphic orbit problem for polynomial algebras

It is proved that every endomorphism preserving the automorphic orbit of a non-trivial element of the rank two polynomial algebra over the complex number field is an automorphism. © 2007 Elsevier Inc. All rights reserved.postprin

On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) K1 and K2 over a field k, a k-isomorphism of K1 [t] (K1 (t)) and K2 [t] (K2 (t)) implies a k-isomorphism of K1 and K2. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:. 1. Let K be an affine field over an algebraically closed field k of any characteristic. SupposeK (t) ≃ k (t1, t2, t3), thenK ≃ k (t1, t2) . 2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. LetA = K [x, y, z, w] / M be the coordinate ring of M. SupposeA [t] ≃ k [x1, x2, x3, x4], thenfrac (A) ≃ k (x1, x2, x3), wherefrac (A) is the field of fractions of A. In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171]. © 2008 Elsevier Inc. All rights reserved.preprin

The strong anick conjecture is true

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K〉x, y, z〈 over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K〈x, y, z〉. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K〈x, y, z〉. We also find a large new class of wild automorphisms of K〈x, y, z〉 which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results. © European Mathematical Society 2007.postprin

Embeddings of hypersurfaces in affine spaces

In this paper, we address the following two general problems: given two algebraic varieties in Cn, find out whether or not they are (1) isomorphic and (2) equivalent under an automorphism of Cn. Although a complete solution of either of those problems is out of the question at this time, we give here some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one and use it to construct numerous examples of isomorphic but inequivalent algebraic varieties in Cn. Among other things, we establish the following interesting fact: for isomorphic hypersurfaces p(x1,...,xn)=0 and q(x1,...,xn)=0, the number of zeros of grad(p) might be different from that of grad(q). This implies, in particular, that, although the fibers p=0 and q=0 are isomorphic, there are some other fibers p=c and q=c which are not. We construct examples like this for any n≥2. © 2001 Academic Press.postprin

Equivalence of polynomials under automorphisms of K [x, y]

Let K [x, y] be the algebra of polynomials in two variables over an arbitrary field K. We show that if the maximum of the x- and y-degrees of a given polynomial p (x, y) cannot be decreased by a single triangular or linear automorphism of K [x, y], then it cannot be decreased by any automorphism of K [x, y]. If K is an algebraically closed constructible field, this result yields an algorithm for deciding whether or not two polynomials p, q ∈ K [x, y] are equivalent under an automorphism of K [x, y]. We also show that if there is an automorphism of K [x, y] taking p to q, then it is "almost" unique. More precisely: if an automorphism α of K [x, y] is not conjugate to a triangular or linear automorphism, then any polynomial invariant (or even semiinvariant) under α is a constant. © 2006 Elsevier Ltd. All rights reserved.preprin

Bilinear probabilistic principal component analysis

Probabilistic principal component analysis (PPCA) is a popular linear latent variable model for performing dimension reduction on 1-D data in a probabilistic manner. However, when used on 2-D data such as images, PPCA suffers from the curse of dimensionality due to the subsequently large number of model parameters. To overcome this problem, we propose in this paper a novel probabilistic model on 2-D data called bilinear PPCA (BPPCA). This allows the establishment of a closer tie between BPPCA and its nonprobabilistic counterpart. Moreover, two efficient parameter estimation algorithms for fitting BPPCA are also developed. Experiments on a number of 2-D synthetic and real-world data sets show that BPPCA is more accurate than existing probabilistic and nonprobabilistic dimension reduction methods.published_or_final_versio

Advances in Alzheimer's disease: from bench to bedside

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Hmong Adults Self-Rated Oral Health: A Pilot Study

Since 1975, the Hmong refugee population in the U.S. has increased over 200%. However, little is known about their dental needs or self-rated oral health (SROH). The study aims were to: (1) describe the SROH, self-rated general health (SRGH), and use of dental/physician services; and (2) identify the factors associated with SROH among Hmong adults. A cross-sectional study design with locating sampling methodology was used. Oral health questionnaire was administered to assess SROH and SRGH, past dental and physician visits, and language preference. One hundred twenty adults aged 18–50+ were recruited and 118 had useable information. Of these, 49% rated their oral health as poor/fair and 30% rated their general health as poor/fair. Thirty-nine percent reported that they did not have a regular source of dental care, 46% rated their access to dental care as poor/fair, 43% visited a dentist and 66% visited a physician within the past 12 months. Bivariate analyses demonstrated that access to dental care, past dental visits, age and SRGH were significantly associated with SROH (P \u3c 0.05). Multivariate analyses demonstrated a strong association between access to dental care and good/excellent SROH. About half of Hmong adults rated their oral health and access to dental care as poor. Dental insurance, access to dental care, past preventive dental/physician visits and SRGH were associated with SROH

Learning Shape Priors for Single-View 3D Completion and Reconstruction

The problem of single-view 3D shape completion or reconstruction is challenging, because among the many possible shapes that explain an observation, most are implausible and do not correspond to natural objects. Recent research in the field has tackled this problem by exploiting the expressiveness of deep convolutional networks. In fact, there is another level of ambiguity that is often overlooked: among plausible shapes, there are still multiple shapes that fit the 2D image equally well; i.e., the ground truth shape is non-deterministic given a single-view input. Existing fully supervised approaches fail to address this issue, and often produce blurry mean shapes with smooth surfaces but no fine details. In this paper, we propose ShapeHD, pushing the limit of single-view shape completion and reconstruction by integrating deep generative models with adversarially learned shape priors. The learned priors serve as a regularizer, penalizing the model only if its output is unrealistic, not if it deviates from the ground truth. Our design thus overcomes both levels of ambiguity aforementioned. Experiments demonstrate that ShapeHD outperforms state of the art by a large margin in both shape completion and shape reconstruction on multiple real datasets.Comment: ECCV 2018. The first two authors contributed equally to this work. Project page: http://shapehd.csail.mit.edu