# Numerical curves and their applications to algebraic curves

Studia Mathematica (1996)

• Volume: 121, Issue: 3, page 249-275
• ISSN: 0039-3223

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## Abstract

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Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme $N={n}_{1},...,{n}_{s};n$, i.e., the sequence of multiplicities of nodes associated with the degree of interpolating polynomials, is considered. We continue the investigation of canonical decomposition of schemes and define so called maximal schemes. Some numerical results concerning the factorization of schemes are established. This leads to determination of irreducibility or to finding the (exact) number of components of algebraic curves as well as to the characterization of all singular points of a wide family of algebraic curves. Also, the Hilbert function of schemes is discussed. At the end, the problem of regularity of schemes depending on the number of interpolation conditions is considered.

## How to cite

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Gevorgian, H., Hakopian, H., and Sahakian, A.. "Numerical curves and their applications to algebraic curves." Studia Mathematica 121.3 (1996): 249-275. <http://eudml.org/doc/216355>.

@article{Gevorgian1996,
abstract = {Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme $N=\{n_1,..., n_s;n\}$, i.e., the sequence of multiplicities of nodes associated with the degree of interpolating polynomials, is considered. We continue the investigation of canonical decomposition of schemes and define so called maximal schemes. Some numerical results concerning the factorization of schemes are established. This leads to determination of irreducibility or to finding the (exact) number of components of algebraic curves as well as to the characterization of all singular points of a wide family of algebraic curves. Also, the Hilbert function of schemes is discussed. At the end, the problem of regularity of schemes depending on the number of interpolation conditions is considered.},
author = {Gevorgian, H., Hakopian, H., Sahakian, A.},
journal = {Studia Mathematica},
keywords = {Hermite interpolation; bivariate algebraic polynomials; algebraic curves; scheme},
language = {eng},
number = {3},
pages = {249-275},
title = {Numerical curves and their applications to algebraic curves},
url = {http://eudml.org/doc/216355},
volume = {121},
year = {1996},
}

TY - JOUR
AU - Gevorgian, H.
AU - Hakopian, H.
AU - Sahakian, A.
TI - Numerical curves and their applications to algebraic curves
JO - Studia Mathematica
PY - 1996
VL - 121
IS - 3
SP - 249
EP - 275
AB - Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme $N={n_1,..., n_s;n}$, i.e., the sequence of multiplicities of nodes associated with the degree of interpolating polynomials, is considered. We continue the investigation of canonical decomposition of schemes and define so called maximal schemes. Some numerical results concerning the factorization of schemes are established. This leads to determination of irreducibility or to finding the (exact) number of components of algebraic curves as well as to the characterization of all singular points of a wide family of algebraic curves. Also, the Hilbert function of schemes is discussed. At the end, the problem of regularity of schemes depending on the number of interpolation conditions is considered.
LA - eng
KW - Hermite interpolation; bivariate algebraic polynomials; algebraic curves; scheme
UR - http://eudml.org/doc/216355
ER -

## References

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1. [GHS90] H. Gevorgian, H. Hakopian and A. Sahakian, On bivariate Hermite interpolation, Mat. Zametki 48 (1990), 137-139 (in Russian).
2. [GHS92] H. Gevorgian, H. Hakopian and A. Sahakian, On the bivariate polynomial interpolation, Mat. Sb. 183 (1992), 111-126 (in Russian); English transl.: Russian Acad. Sci. Sb. Math. 76 (1993), 211-223.
3. [GHS92b] H. Gevorgian, H. Hakopian and A. Sahakian, On the bivariate Hermite interpolation problem, Constr. Approx. 11 (1995), 23-35. Zbl0823.41004
4. [GHS93] H. Gevorgian, H. Hakopian and A. Sahakian, Bivariate Hermite interpolation and numerical curves, in: Proc. of the International Conference on Open Problems in Approximation Theory, Voneshta Voda, 1993, 85-89.
5. [GHS95] H. Gevorgian, H. Hakopian and A. Sahakian, Bivariate Hermite interpolation and numerical curves, J. Approx. Theory 85 (1996), 297-317. Zbl0857.41001
6. [H94] H. Hakopian, Multivariate Polynomial Interpolation, Habilitation survey, Mathematical Institute of Polish Acad. Sci., Warszawa, 1994.
7. [JS91] R-Q. Jia and A. Sharma, Solvability of some multivariate interpolation problems, J. Reine Angew. Math. 421 (1991), 73-81. Zbl0735.41002
8. [LL84] G. G. Lorentz and R. A. Lorentz, Multivariate interpolation, in: Lecture Notes in Math. 1105, Springer, Berlin, 1984, 136-144. Zbl0566.41001
9. [P92] S. H. Paskov, Singularity of bivariate interpolation, J. Approx. Theory 75 (1992), 50-67. Zbl0770.41006
10. [W50] R. Walker, Algebraic Curves, Princeton Univ. Press, Princeton, N.J., 1950.

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