Project's information

Project's title Some problems in Singularity Theory and Applications
Project’s code ĐLTE00.01/21-22
Research hosting institution Institute of Mathematics
Project leader’s name Dinh Si Tiep
Project duration 01/01/2021 - 30/06/2023
Classify Grade B
Goal and objectives of the project
-  Subject 1: Study the limits of the rational function f(x)/g(x) at a point.
- Subject 2: Study the set of exceptional directions, the closure of the image of definable closed set by continuous definable mapping, the Lipschitz continuity of tangent directions at infinity, some results of classical analysis for continuous definable mapping.
  - Subject 3: Compare motivic and non-Archimedean Milnor fibers.
Main results
- Theoretical results: 
+ Subject 1: Provide a formula for calculating the Lojasiewicz exponent for comparing the rate of growth of two real polynomial functions at a common isolated zero by showing that this exponent is attained on the tangency curve. Then show that this Lojasiewicz exponent can be computed via truncated Puiseux series of tangency curve. This is related to the existence of the limit of a rational function in two real variables.
+ Subject 2:
• Describe Nash fibers and exceptional directions at a point on a definable set (in a given o-minimum structure).
• Provide sufficient conditions for the image of a semi-algebraic set by a continuous semi-algebraic mapping to be closed.
• Prove that the closedness of the image of a semi-algebraic set by a continuous semi-algebraic mapping is a generic property.
• Show that when the value of a given polynomial function f varies, the set of tangent directions at infinity of the corresponding fiber of f and its Hausdorff (n-2)-dimensional measure varies locally Lipschitz continuously outside the set of asymptotic critical values of f.
• Give some results of classical analysis for definable continuous mappings (non-smooth Sard theorem, mean value theorem, implicit function theorem, inverse function theorem,...).
+ Subject 3: Give a formula for the motivic Milnor fiber of a non-degenerate polynomial in terms of its Newton polygon.
Novelty and actuality and scientific meaningfulness of the results

- Improve computational time in checking for existence and computing limit of rational functions in two real variables.
- Describe Nash fibers and exceptional directions in a more general context than known cases. Application in the study of closure.
- Prove that the closedness of the image of a semi-algebraic set by a continuous semi-algebraic mapping is a generic property.
- Prove the locally Lipschitz continuity of the tangent directions at infinity of fibers of a polynomial function outside its set of asymptotic critical values of f.
- Give some results of classical analysis for a class of non-smooth mappings (defined continuous mappings).
- Give a relation between the motivic Milnor fibers of a function f with the fibers of the restriction of f on a coordinate hyperplane and give computation on cohomology groups of the contact loci.
Products of the project

- Scientific papers in referred journals (list):
1. Đinh Sĩ Tiệp and Phạm, Tiến-Sơn, Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings, Proc. Amer. Math. Soc. 150 (2022), no. 9, 3663–3673.
2.  Ha, Truong Xuan Duc and Phạm, Tiến-Sơn, Some classical analysis results for continuous definable mappings, J. Math. Anal. Appl. 515 (2022), no. 1, Paper No. 126380, 19 pp.
3. Đinh Sĩ Tiệp and Phạm Tiến Sơn, Lipschitz continuity of tangent directions at infinity, Bull. Sci. Math. 182 (2023), Paper No. 103223, 27 pp.
- Other products (if applicable):
1. Lê Quý Thường and Nguyễn Tất Thắng, Geometry of nondegenerate polynomials: motivic nearby cycles and cohomology of contact loci, Comptes Rendus Mathématique (accepted).
2. A summary of the goals and results of the project.