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A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS

icon-email WENHUA ZHAO

Abstract

In the recent progress [4], [17] and [25], the well-known JC (Jacobian conjecture) ([2], [10]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd order homogeneous differential operators Λ and Λ-nilpotent polynomials P (the polynomials P(z) satisfying ΛmPm=0 for all m⩾1). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on -nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators Λ. This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of Λ-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous Λ-nilpotent polynomials for 2nd order homogeneous full rank differential operators Λ with constant coefficients.

Viện Toán học, Viện Hàn lâm Khoa học và Công nghệ Việt Nam

Địa chỉ: Số 18 đường Hoàng Quốc Việt, quận Cầu Giấy, Hà Nội

Điện thoại: 024 37563474 - Fax: 024 37564303

Email: vientoan@math.ac.vn

Viện nghiên cứu cao cấp về toán Hội Toán học Mỹ MathScinet Thư viện số (VAST) Arxiv
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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