The implicit function theorem and free algebraic sets
نویسندگان
چکیده
منابع مشابه
The implicit function theorem and free algebraic sets
We prove an implicit function theorem for non-commutative functions. We use this to show that if p(X, Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p(X, Y ) = 0 will commute with X.
متن کاملImplicit function theorem over free groups
We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the Implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lift...
متن کاملThe Implicit Function Theorem and Implicit Parametrizations∗
We discuss a differential equations treatment of the implicit functions problem. Our approach allows a precise and complete description of the solution, of continuity and differentiability properties. The critical case is also considered. The investigation is devoted to dimension two and three, but extensions to higher dimension are possible. MSC: 26B10, 34A12, 53A05. keywords: implicit functio...
متن کاملThe Contraction Mapping Theorem and the Implicit Function Theorem
denote the open ball of radius a centred on the origin in IR. If the function ~g : Ba → IR d obeys there is a constant G < 1 such that ‖~g(~x)− ~g(~y)‖ ≤ G ‖~x− ~y‖ for all ~x, ~y ∈ Ba (H1) ‖~g(~0)‖ < (1−G)a (H2) then the equation ~x = ~g(~x) has exactly one solution. Discussion of hypothesis (H1): Hypothesis (H1) is responsible for the word “Contraction” in the name of the theorem. Because G <...
متن کاملComputability and the Implicit Function Theorem
We prove computable versions of the Implicit Function Theorem in the single and multivariable cases. We use Type Two Effectivity as our foundation.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2015
ISSN: 0002-9947,1088-6850
DOI: 10.1090/tran/6546