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<p>In this note is given an algebraic solution to the problem 1997-6 proposed by D. A. Panov in the list of Arnold’s problems (in Arnold’s Problems. Springer, Berlin, 2024). In particular, it is shown that there does not exist a real polynomial function <i>f</i> on the real euclidean plane, whose Hessian is positive in an open set bordered by smooth connected curve, and the parabolic curve of the graph of <i>f</i> has only one special parabolic point with index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Besides, we find conditions on <i>f</i> so that its graph has more special parabolic points with index <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> than with index +1.</p>

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On the non-existence of certain real algebraic surfaces

  • 期刊论文

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  • 发表日期：

    2026年3月30日

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  • Miguel Angel Guadarrama-García

摘要

In this note is given an algebraic solution to the problem 1997-6 proposed by D. A. Panov in the list of Arnold’s problems (in Arnold’s Problems. Springer, Berlin, 2024). In particular, it is shown that there does not exist a real polynomial function f on the real euclidean plane, whose Hessian is positive in an open set bordered by smooth connected curve, and the parabolic curve of the graph of f has only one special parabolic point with index \(+1\) + 1 . Besides, we find conditions on f so that its graph has more special parabolic points with index \(-1\) - 1 than with index +1.