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In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials Fm,n,1(x) and Fm,n,2(x). Then, he defined Am(n,k) and Bm(n,k) to be the polynomials satisfying Fm,n,1(x)=k=0nAm(n,k)xn?k(x+1)k and Fm,n,1(x)=k=0nBm(n,k)xn?k(x+1)k. In this paper, we give a combinatorial interpretation of the coefficients of Am+1(n,k) and prove a symmetry of the coefficients, i.e., [ms]Am+1(n,k)=[mn?s]Am+1(n,n?k). We give a combinatorial interpretation of Bm+1(n,k) and prove that Bm+1(n,n?1) is a polynomial in m with non-negative integer coefficients. We also prove that if n6 then all coefficients of Bm+1(n,n?2) except the coefficient of mn?1 are non-negative integers. For all n, the coefficient of mn?1 in Bm+1(n,n?2) is ?(n?1), and when n5 some other coefficients of Bm+1(n,n?2) are also negative.  相似文献   

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This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if {fj}j=1n2 are analytic trigonometric polynomials without common zero in the finite complex plane ? then there are analytic trigonometric polynomials {gj}j=1n2 obeying j=1n2fjgj=1 in ?, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on ?.  相似文献   

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In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, Mpc(n), with degrees 2, 3, 4, and 5. More concretely, Mpc(2)13, Mpc(3)26, Mpc(4)40, and Mpc(5)58. The computations use parallelization algorithms.  相似文献   

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We consider the irreducibility of polynomial Ln(α)(x) where α is a negative integer. We observe that the constant term of Ln(α)(x) vanishes if and only if n|α|=?α. Therefore we assume that α=?n?s?1 where s is a non-negative integer. Let g(x)=(?1)nLn(?n?s?1)(x)=j=0najxjj! and more general polynomial, let G(x)=j=0najbjxjj! where bj with 0jn are integers such that |b0|=|bn|=1. Schur was the first to prove the irreducibility of g(x) for s=0. It has been proved that g(x) is irreducible for 0s60. In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either G(x) is irreducible or G(x) is linear factor times irreducible polynomial. This is a consequence of the estimate s>1.9k whenever G(x) has a factor of degree k2 and (n,k,s)(10,5,4). This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.  相似文献   

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Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set C. It is shown that, for each m3, every real number in the unit interval [0,1] is the sum x1m+x2m+?+xnm with each xj in C and some n6m. Furthermore, every real number x in the interval [0,8] can be written as x=x13+x23+?+x83, the sum of eight cubic powers with each xj in C. Another Cantor set C×C is also considered. More specifically, when C×C is embedded into the complex plane ?, the Waring–Hilbert problem on C×C has a positive answer for powers less than or equal to 4.  相似文献   

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《Discrete Mathematics》2019,342(4):1089-1097
Given integers pq>1, a family of sets satisfies the (p,q) property if among any p members of it some q intersect. We prove that for any fixed integer constants pq>1, a family of d-intervals satisfying the (p,q) property can be pierced by O(dqq1) points, with constants depending only on p and q. This extends results of Tardos, Kaiser and Alon for the case q=2, and of Kaiser and Rabinovich for the case p=q=log2(d+2). We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consisting of at most d connected components, extending results of Alon for the case q=2. Finally, we prove an upper bound of O(d1p1) on the fractional piercing number in families of d-intervals satisfying the (p,p) property, and show that this bound is asymptotically sharp.  相似文献   

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