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: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of i ...
on the absolute dimension of a differential algebraic variety in terms of its defining equations. This is one of Kolchin's Problems. The inequality is the differential algebraic analog of
Bézout's theorem In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the de ...
in affine space. Although first formulated by Jacobi, In 1936
Joseph Ritt Joseph Fels Ritt (August 23, 1893 – January 5, 1951) was an American mathematician at Columbia University in the early 20th century. He was born and died in New York. Biography After beginning his undergraduate studies at City College of Ne ...
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 mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients ...
). 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 (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s. A
mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
of the inequality has been open since 1936.


Statement

Let (K,\partial) be a
differential field In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebra, algebraic objects in view of deriving properties of differential equations ...
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