The existence of a linear manifold in a kernel of a complex 
polynomial functional on a finite dimension linear space (in Ukrainain)

A. Zagorodnyuk

We prove the following result. For any positive integers $d_1,\dots,d_s$ there exists a monotone increasing function $\Psi (d_1,\dots,d_s,m)\colon {\Bbb Z}_+\to {\Bbb Z}_+$ such that if $\Psi (d_1,\dots ,d_s,m)$ $=$ $n$ then for each homogeneous polynomial functionals $p_1,p_2,\dots,p_s$ of degrees $d_1,\dots,d_s$ on a complex linear spase $X$, $\dim X=n$, there exists a linear subspace $V\subset X$, $\dim V=m$ such that $V\subset \bigcap _{i=1}^s\ker p_i$. Moreover, if $\dim X=\infty $, then $\dim V=\infty $.

1. Bochnak J., Sisiak J. Polynomials and multilinear mappings in topological vector spaces // Stud. Math. 1971. V.39. P.77–112.

2. Dineen S. Complex analysis in locally convex spaces. — Amsterdam et al.: North-Holland, 1981. 492 p.

3. Hyers D.H. Polynomial operators // Topics in Mathematical Analysis. 1989. P.410–444.

4. Рид. М. Алгебраическая геометрия для всех. - М. :Мирб 1991 - 152 с. 5. Plichko A., Zagorodnyuk A. On automatic continuity and three problems of "The Scottish Book" concerning the boundedness of polynomial functionals // J. Math. Anal. and Appl. (to appear).

	The existence of a linear manifold in a kernel of a complex polynomial functional on a finite dimension linear space (in Ukrainain)
Author	A. Zagorodnyuk Department of Nonlinear Mathematical Analysis, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the NAS of Ukraine, 3b Naukova St., Lviv, Ukraine
Abstract	We prove the following result. For any positive integers $d_1,\dots,d_s$ there exists a monotone increasing function $\Psi (d_1,\dots,d_s,m)\colon {\Bbb Z}_+\to {\Bbb Z}_+$ such that if $\Psi (d_1,\dots ,d_s,m)$ $=$ $n$ then for each homogeneous polynomial functionals $p_1,p_2,\dots,p_s$ of degrees $d_1,\dots,d_s$ on a complex linear spase $X$, $\dim X=n$, there exists a linear subspace $V\subset X$, $\dim V=m$ such that $V\subset \bigcap _{i=1}^s\ker p_i$. Moreover, if $\dim X=\infty $, then $\dim V=\infty $.
Keywords	homogeneous polynomial functionals, degrees, complex linear space, linear subspace
DOI	doi:10.30970/ms.8.1.115-118
Reference	1. Bochnak J., Sisiak J. Polynomials and multilinear mappings in topological vector spaces // Stud. Math. 1971. V.39. P.77–112. 2. Dineen S. Complex analysis in locally convex spaces. — Amsterdam et al.: North-Holland, 1981. 492 p. 3. Hyers D.H. Polynomial operators // Topics in Mathematical Analysis. 1989. P.410–444. 4. Рид. М. Алгебраическая геометрия для всех. - М. :Мирб 1991 - 152 с. 5. Plichko A., Zagorodnyuk A. On automatic continuity and three problems of "The Scottish Book" concerning the boundedness of polynomial functionals // J. Math. Anal. and Appl. (to appear).
Pages	115-118
Volume	8
Issue	1
Year	1997
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The existence of a linear manifold in a kernel of a complex polynomial functional on a finite dimension linear space (in Ukrainain)