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Geometric properties of the lattice of polynomials with integer coefficients

Lipnicki Artur, Śmietański Marek J.

Opuscula Mathematica

2024

Vol. 44, no. 4

565--585

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let r, n be positive integers with n ≥ 6r. Let Pn ∩Mr be the space of polynomials of degree at most n on [0, 1] with integer coefficients such that P(k)(0)/k! and P(k)(1)/k! are integers for k = 0, . . . , r − 1 and let PZn ∩Mr be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of PZn ∩Mr from Pn ∩Mr in L2(0, 1). We give rather precise quantitative estimations for successive minima of PZn in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in Pn ∩Mr.

About a certain relation between two polynomials of the same degree

Piwowarczyk T.

Archives of Control Sciences

2011

Vol. 21, no. 1

85-103

This article presents several different methods for solving the problem of how to find a certain relation defined in chapter 2. The first method deals with the identities known in the theory of symmetric polynomials as the elements of a certain vector space. The second method is designed around the matrix transformations between symmetric polynomials. The third method is designed around the property of a linear operator and its characteristic polynomial. The fourth method is designed in the area of complex numbers, and introduces the multiplication group of 'complex roots of one'. Significant improvement in the third and fourth method is made by introducing so called 'block method'. It facilitates all calculations by making them much shorter. The article ends with an example showing symmetry and regularity of all procedures. Finally, the article shows how to solve the problem for any degree n of the polynomial, and for any degree k. At the end of the paper solutions for n < 5 and k < 5 are tabulated.

Stability of the convex combination of polynomials

Góra M.

Control and Cybernetics

2007

Vol. 36, no 2

425-442

In this paper, we consider the convex combination of polynomials. We provide a necessary and sufficient condition for Hurwitz stability of the convex combination of m real polynomials (m ≥ 3) whose degrees may be different and both necessary, and necessary and sufficient conditions for Hurwitz and Schur stability of the convex combination of two complex polynomials. We show also that the convex combination of two polynomials whose degrees are respectively odd and even, is never Schur stable. We give a few examples completing the results.