 

Raúl E. Curto and
Seonguk Yoo
A moment theoretic approach to estimate the cardinality of certain algebraic varieties
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Published: 
February 4, 2022. 
Keywords: 
Flat Extension Theorem; planar algebraic curves; truncated moment problems;
Bézout's Theorem. 
Subject: 
Primary 47A57, 44A60, 14H50; Secondary 1504, 12A10. 


Abstract
For any positive integer n we consider the algebraic variety V obtained by intersecting n+1 algebraic curves of degree n in R^{2}, when the leading terms of the associated bivariate polynomials are all different. We provide a new proof, based on the Flat Extension Theorem from the theory of truncated moment problems, that the cardinality of V cannot exceed n(n+1)/2. In some instances, this provides a slightly better estimate than the one given by Bézout's Theorem. Our main result contributes to the growing literature on the interplay between linear algebra, operator theory, and real algebraic geometry.


Acknowledgements
The second named author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF2020R1F1A1A01070552).


Author information
Raúl E. Curto:
Department of Mathematics
University of Iowa
Iowa City, Iowa 52242, USA
raulcurto@uiowa.edu
Seonguk Yoo:
Department of Mathematics Education and RINS
Gyeongsang National University
Jinju, Republic of Korea 52828
seyoo@gnu.ac.kr

