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The Jacobian Conjecture and Kernel Conjecture | ||
| journal of al-qadisiyah for pure science(quarterly) | ||
| Article 1, Volume 9, Issue 1, 2009, Pages 61-62 PDF (0 K) | ||
| Author | ||
| A-Hamid A-Hussain Ali | ||
| Abstract | ||
| In this paper we study the Jacobian conjecture from the viewpoint of derivations and relate it to a conjecture about the kernel of a derivation. 1. Derivations and the Jacobian condition. The aim of this section is to study the Jacobian conjecture by means of derivations therefore we reformulate the Jacobian conjecture in terms of the kernel of a special derivation. Let be a polynomial map satisfying the Jacobian condition , det * .To such a map we associate an n-tuple of derivations on denoted by as follows ... 2- The Jacobian conjecture and kernel conjecture Let be polynomial map with det * and assume that the jacobian conjecture is true . Then Hence ker This leads us to Conjecture (kernel conjecture KC(n)):if det * ,then . The observation above then states that JC(n) implies KC(n), for all . Conversely we have Proposition : KC(n+1) implies JC(n) for all . proof: Let with det *. | ||
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