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<p>M.&#xa0;Janet in 1921 conjectured that an analytic solution to systems of <i>n</i> consistent <i>m</i>-partial differential equations of <i>n</i>&#xa0;unknown functions must contain at least one arbitrary function of <i>k</i> variables, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\ge m-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. E.&#xa0;Kolchin at the Moscow International Congress in 1966 formulated an algebraic version of this conjecture. In the case of linear systems, it was proven by J.&#xa0;Johnson in 1978, but for nonlinear systems the question is still open. This paper shows that the generalized Janet conjecture does not hold for the intersection of <i>n</i> differential hyperspaces in the case of any number of derivations <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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ON THE CONJECTURE OF M. JANET FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • M. V. Kondratieva

摘要

M. Janet in 1921 conjectured that an analytic solution to systems of n consistent m-partial differential equations of n unknown functions must contain at least one arbitrary function of k variables, \(k\ge m-1\) k ≥ m - 1 . E. Kolchin at the Moscow International Congress in 1966 formulated an algebraic version of this conjecture. In the case of linear systems, it was proven by J. Johnson in 1978, but for nonlinear systems the question is still open. This paper shows that the generalized Janet conjecture does not hold for the intersection of n differential hyperspaces in the case of any number of derivations \(m>0\) m > 0 .