The perfectoid Tate algebra has uncountable Krull dimension

Let $K$ be a perfectoid field with pseudo-uniformizer $\pi$. We adapt an argument of Du to show that the perfectoid Tate algebra $K\langle x^{1 / p^{\infty}} \rangle$ has an uncountable chain of distinct prime ideals. First, we conceptualize Du's argument, defining the notion of a 'Newton polygon formalism' on a ring. We prove a version of Du's theorem in the prescence of a sufficiently nondiscrete Newton polygon formalism. Then, we apply our framework to the perfectoid Tate algebra via a "nonstandard" Newton polygon formalism (roughly, the roles of the series variable $x$ and the pseudo-uniformizer $\pi$ are switched). We conclude a similar statement for multivatiate perfectoid Tate algebras using the one-variable case. We also answer a question of Heitmann, showing that if $R$ is a complete local noetherian domain of mixed characteristic $(0,p)$, the $p$-adic completion of it's absolute integral closure $R^{+}$ has uncountable Krull dimension.Comment: 15 pages, 2 figures, comments welcom

Semilocal analysis of equations with smooth operators

Recent work on semilocal analysis of nonlinear operator equations is informally reviewed. A refined version of the Kantorovich theorem for Newton's method, with new error bounds, is presented. Related topics are briefly surveyed

Semilocal Convergence of the Extension of Chun's Method

[EN] In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun's iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Frechet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and FONDOCYT 027-2018 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Martínez Molada, E.; Torregrosa Sánchez, JR.; Vassileva, MP. (2021). Semilocal Convergence of the Extension of Chun's Method. Axioms. 10(3):1-11. https://doi.org/10.3390/axioms10030161S11110

Iterative methods with memory for solving systems of nonlinear equations using a second order approximation

[EN] Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089, and FONDOCYT 2016-2017-212 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Torregrosa Sánchez, JR.; Vassileva, MP. (2019). Iterative methods with memory for solving systems of nonlinear equations using a second order approximation. Mathematics. 7(11):1-12. https://doi.org/10.3390/math7111069S112711Soleymani, F., Lotfi, T., Tavakoli, E., & Khaksar Haghani, F. (2015). Several iterative methods with memory using self-accelerators. Applied Mathematics and Computation, 254, 452-458. doi:10.1016/j.amc.2015.01.045Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048Potra, F. A. (1982). An error analysis for the secant method. Numerische Mathematik, 38(3), 427-445. doi:10.1007/bf01396443Fatou, P. (1919). Sur les équations fonctionnelles. Bulletin de la Société mathématique de France, 2, 161-271. doi:10.24033/bsmf.998Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Campos, B., Cordero, A., Torregrosa, J. R., & Vindel, P. (2015). A multidimensional dynamical approach to iterative methods with memory. Applied Mathematics and Computation, 271, 701-715. doi:10.1016/j.amc.2015.09.056Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015

Extending the solvability of equations using secant-type methods in Banach space

We extend the solvability of equations dened on a Banach space using numerically ecient secant-type methods. The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sucient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational eort, since the new Lipschitz constants are special cases of the old ones

Convergence analysis for the two-step Newton method of order four

We provide a tighter than before convergence analysis for the two-step Newton method of order four using recurrent functions. Numerical examples are also provided in this study