scholarly journals Essentially finite generation of valuation rings in terms of classical invariants

2020 ◽  
Author(s):  
Steven Dale Cutkosky ◽  
Josnei Novacoski
Keyword(s):  
Finite Generation ◽  
Valuation Rings

scholarly journals On three-variable expanders over finite valuation rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Le Quang Ham ◽  
Nguyen Van The ◽  
Phuc D. Tran ◽  
Le Anh Vinh
Keyword(s):  
Valuation Ring ◽  
Product Type ◽  
Quadratic Polynomial ◽  
Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

scholarly journals Valuation rings are derived splinters

2021 ◽  
Author(s):  
Benjamin Antieau ◽  
Rankeya Datta
Keyword(s):  
Valuation Rings

scholarly journals Algebraic series and valuation rings over nonclosed fields

2008 ◽  
Vol 212 (8) ◽  
pp. 1996-2010 ◽  
Author(s):  
Steven Dale Cutkosky ◽  
Olga Kashcheyeva
Keyword(s):  
Valuation Rings

Finitely generated modules over valuation rings

1984 ◽  
Vol 12 (15) ◽  
pp. 1795-1812 ◽  
Author(s):  
Luigi Salce ◽  
Paolo Zanardo
Keyword(s):  
Finitely Generated ◽  
Valuation Rings ◽  
Finitely Generated Modules

A note on Itoh (e)-valuation rings of an ideal

2018 ◽  
Vol 498 ◽  
pp. 47-70
Author(s):  
Youngsu Kim ◽  
Louis J. Ratliff ◽  
David E. Rush
Keyword(s):  
Valuation Rings

scholarly journals Valuation rings in finite-dimensional division algebras

1989 ◽  
Vol 120 (1) ◽  
pp. 90-99 ◽  
Author(s):  
H.H Brungs ◽  
Joachim Gräter
Keyword(s):  
Division Algebras ◽  
Finite Dimensional ◽  
Valuation Rings

scholarly journals Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

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