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Research Paper | Mathematics | India | Volume 13 Issue 5, May 2024 | Rating: 5.6 / 10
Automorphisms of Afne Spaces: A Proof of Bijectivity via Jacobian Determinant
Abstract: In this research, we delve into the intricate interplay between algebraic structures and geometric properties in the context of morphisms in affine spaces. Specifically, we focus on a morphism ϕ : A^n -> A^n defined by polynomials (f1, f2, ...., fn) and investigate the conditions under which ϕ becomes an automorphism. We provide a comprehensive interpretation of the Jacobian matrix J(ϕ), a fundamental tool in understanding local behavior, emphasizing its role in capturing the local sensitivity of each component of ϕ to changes in the variables. We investigate the implications of having a non-zero constant Jacobian determinant, establishing its pivotal role in ensuring both injectivity and surjectivity of ϕ. The connectedness of the affine space A^n proves to be a crucial factor, allowing us to combine local inverses obtained from the Implicit Function Theorem to construct a global inverse for ϕ. Beyond the theoretical framework, our research opens up exciting avenues for future exploration. We propose directions for generalizing these results to more abstract spaces, investigating morphisms defined by non-polynomial functions, and exploring applications in computer science, cryptography, and algebraic geometry. In conclusion, our study contributes to the rich tapestry of mathematical structures, revealing the profound connections between algebraic properties and geometric structures in the realm of morphisms and automorphisms.
Keywords: Automorphisms, Afne Spaces, Jacobian Matrix, Jacobian Determinant, Morphisms Defined by Polynomials, Implicit Function Theorem, Local Invertibility, Connectedness of Spaces, Algebraic Structures, Geometric Interpretation
Edition: Volume 13 Issue 5, May 2024,
Pages: 576 - 585