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On Roots of Polynomials and Algebraically Closed Fields

100%
Schwarzweller C.
Formalized Mathematics
|
2017
|
tom 25
|
nr 3
185-195
EN
In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].
2
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Banach spaces in which all multilinear forms are weakly sequentially continuous

100%
Castillo J. M. F., García R., Gonzalo R.
Studia Mathematica
|
1999
|
tom 136
|
nr 2
121-145
EN
We solve several problems in the theory of polynomials in Banach spaces. (i) There exist Banach spaces without the Dunford-Pettis property and without upper p-estimates in which all multilinear forms are weakly sequentially continuous: some Lorentz sequence spaces, their natural preduals and, most notably, the dual of Schreier's space. (ii) There exist Banach spaces X without the Dunford-Pettis property such that all multilinear forms on X and X* are weakly sequentially continuous; this gives an answer to a question of Dimant and Zalduendo [20]. (iii) The sum of two polynomially null sequences need not be polynomially null; this answers a question of Biström, Jaramillo and Lindström [8] and also of González and Gutiérrez [23]. (iv), (v) The absolutely convex closed hull of a pw-compact set need not be pw-compact; the projective tensor product of two polynomially null sequences need not be a polynomially null sequence. This answers two questions of González and Gutiérrez [23]. (vi) There exists a Banach space without property (P); this answers a question of Aron, Choi and Llavona [5].
3
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On Subnomials

100%
Ziobro R.
Formalized Mathematics
|
2016
|
tom 24
|
nr 4
261-273
EN
While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer [12] observes “It is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest – mortgages, annuities, etc.” It’s worth noticing that the first issue involves a simple arithmetical division of 99...9 by 9. While the above notion seems not have changed over the last 50 years, it reflects only a special case of a broader class of problems involving two variables. It seems strange, that while binomial formula is discussed and studied widely [7], [8], little research is done on its counterpart with all coefficients equal to one, which we will call here the subnomial. The study focuses on its basic properties and applies it to some simple problems usually proven by induction [6].
4
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High order representation formulas and embedding theorems on stratified groups and generalizations

75%
Lu G., Wheeden R. L.
Studia Mathematica
|
2000
|
tom 142
|
nr 2
101-133
EN
We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^1$ to $L^1$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^1$ to $L^1$ Poincaré inequalities involving high order derivatives are known to hold. We apply the formulas to derive embedding theorems and potential type inequalities involving high order derivatives.
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