The Jacobi Bound Problem
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The Jacobi Bound Problem concerns the veracity of Jacobi's inequality which is an
inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
on the absolute dimension of a differential algebraic variety in terms of its defining equations. The inequality is the differential algebraic analog of Bezout's theorem in affine space. Although first formulated by Jacobi, In 1936 Joseph Ritt recognized the problem as non-rigorous in that Jacobi didn't even have a rigorous notion of absolute dimension (Jacobi and Ritt used the term "order" - which Ritt first gave a rigorous definition for using the notion of
transcendence degree In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
). Intuitively, the absolute dimension is the number of constants of integration required to specify a solution of a system of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s. A
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
of the inequality has been open since 1936.


Statement

Let (K,\partial) be a
differential field In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
of characteristic zero and consider \Gamma a differential algebraic variety determined by the vanishing of differential polynomials u_1,\ldots,u_n \in K _1,\ldots,x_n . If \Gamma_1 is an irreducible component of \Gamma of finite absolute dimension then a(\Gamma_1) \leq J(u_1,u_2,\ldots,u_n). In the above display J(u_1,u_2,\ldots,u_n) is the *jacobi number*. It is defined to be \max_ \sum_^n \operatorname_^(u_) .


References

* * * Unsolved problems in mathematics Differential algebra {{DEFAULTSORT:Jacobi Bound Problem