On the /Lojasiewicz exponent at infinity of real polynomials

Tom 94 / 2008

Ha Huy Vui, Pham Tien Son Annales Polonici Mathematici 94 (2008), 197-208 MSC: Primary 14P10. DOI: 10.4064/ap94-3-1


Let $f \colon {\Bbb R}^n \rightarrow {\Bbb R}$ be a nonconstant polynomial function. Using the information from the “curve of tangency” of $f,$ we provide a method to determine the Łojasiewicz exponent at infinity of $f.$ As a corollary, we give a computational criterion to decide if the /Lojasiewicz exponent at infinity is finite or not. Then we obtain a formula to calculate the set of points at which the polynomial $f$ is not proper. Moreover, a relation between the /Lojasiewicz exponent at infinity of $f$ and the problem of computing the global optimum of $f$ is also established.


  • Ha Huy VuiInstitute of Mathematics
    18, Hoang Quoc Viet Road
    Cau Giay District 10307
    Hanoi, Vietnam
  • Pham Tien SonDepartment of Mathematics
    University of Dalat
    Dalat, Vietnam

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