The neighborhood polynomial of graph $G$ is the generating function for number vertex subsets which vertices have a common neighbor in $G$. In this paper, we investigate behavior under several operations. Specifically, provide an explicit formula obtained from given by attachment. We use result to propose recursive algorithm calculation polynomial. Finally, prove that can be found polynomial-ti...

for every $1leq s< n$, the $s^{th}$ derivative of a polynomial $p(z)$ of degree $n$ is a polynomial $p^{(s)}(z)$ whose degree is $(n-s)$. this paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. besides, our result gives interesting refinements of some well-known results.

in this paper, the concept of canonical representation is proposed to find fuzzy roots of fuzzy polynomial equations. we transform fuzzy polynomial equations to system of crisp polynomial equations, this transformation is perform by using canonical representation based on three parameters value, ambiguity and fuzziness.

D.H. Lehmer found a quadratic polynomial such that 326 is a primitive root for the first 206 primes represented by this polynomial. It is shown that this is related to the class number one problem and prime producing quadratics. More impressive examples in the same spirit are given using recent results on prime producing quadratics. Y. Gallot holds the current record in which 206 is being repla...

The University of Manitoba Sieve Unit is used to find several values of A (> 0) such that the quadratic polynomial x + x + A will have a large asymptotic density of prime values. The Hardy-Littlewood constants which characterize this density are also evaluated.

Abstract. We compute the equation of the 7-secant variety to the Veronese variety (P,O(3)), its degree is 15. This is the last missing invariant in the AlexanderHirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matr...

We focus on a very specific case of the Uniform Boundedness Conjecture, namely, bounding the number of possible c such that the quadratic polynomial φc(z) = z2 + c has quadratic periodic points of some small period. We show that there are infinitely many rational c with quadratic 4-cycles, with all such c completely understood; and only finitely many rational c with quadratic 5-cycles (we conje...

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial ...

D.H. Lehmer found a quadratic polynomial such that 326 is a primitive root for the first 206 primes represented by this polynomial. It is shown that this is related to the class number one problem and prime producing quadratics. More impressive examples in the same spirit are given using recent results on prime producing quadratics. Y. Gallot holds the current record in which 206 is being repla...

We consider polynomial differential systems in the plane with Liouvillian first integrals. It is shown that all such systems have Darbouxian integrating factors, and that the search for such integrals can be reduced to a search for the invariant algebraic curves of the system and their ‘degenerate’ counterparts.