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Register a new userThe number of ideals of $mathbb{Z}[x]$ containing $x(x-alpha)(x-beta)$ with given index
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It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $langle B rangle$ denote the subring of the full matrix ring generated by $B$. Then $langle B rangle$ is a free $mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $langle B rangle$ with given finite index. Thus, the formal Dirichlet series $zeta_{langle B rangle}(s)=sum_{ngeq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $langle B rangle$ with index $n$. In this article we aim to find an explicit form of $zeta_{langle B rangle}(s)$ when $B$ has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, $langle B rangle$ is isomorphic to $mathbb{Z}[x]/m(x)mathbb{Z}[x]$ where $m(x)$ is the minimal polynomial of $B$ over $mathbb{Q}$, and $mathbb{Z}[x]/m(x)mathbb{Z}[x]$ is isomorphic to $mathbb{Z}[x]/m(x+gamma)mathbb{Z}[x]$ for each $gammain mathbb{Z}$. Thus, the problem is reduced to counting the number of ideals of $mathbb{Z}[x]/x(x-alpha)(x-beta)mathbb{Z}[x]$ with given finite index where $0,alpha$ and $beta$ are distinct integers.
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In an earlier article together with Carlos DAndrea [BDKSV2017], we described explicit expressions for the coefficients of the order-$d$ polynomial subresultant of $(x-alpha)^m$ and $(x-beta)^n $ with respect to Bernsteins set of polynomials ${(x-alpha)^j(x-beta)^{d-j}, , 0le jle d}$, for $0le d<min{m, n}$. The current paper further develops the study of these structured polynomials and shows that the coefficients of the subresultants of $(x-alpha)^m$ and $(x-beta)^n$ with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials. The result is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are scalar multiples of Jacobi polynomials up to an affine change of variables.
Let $mathscr{G}_{n,beta}$ be the set of graphs of order $n$ with given matching number $beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_{alpha}(G)=alpha D(G)+A(G)$ is called the $alpha$-spectral radius of $G$. The graphs with maximal $alpha$-spectral radius in $mathscr{G}_{n,beta}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $mathscr{G}_{n,beta}$. More precisely, we generalize the known results on the maximal adjacency spectral radius in $mathscr{G}_{n,beta}$ and the signless Laplacian spectral radius.
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We introduce in a reduced complex space, a new coherent sub-sheaf of the sheaf $omega_{X}^{bullet}$ which has the universal pull-back property for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $Omega_{X}^{bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $Omega_{X}^{bullet}/torsion$. This sheaf $alpha_{X}^{bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic differential forms are locally square-integrable on any $q-$dimensional cycle in $X$ and that the corresponding functions obtained by integration on an analytic family of $q-$cycles are locally bounded and locally continuous on the complement of closed analytic subset.
We characterize semigroups $X$ whose semigroups of filters $varphi(X)$, maximal linked systems $lambda(X)$, linked upfamilies $N_2(X)$, and upfamilies $upsilon(X)$ are commutative.
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