A number field with trivial Pólya group [2] is called a Pólya field. We define "relative Pólya group Po(L/K)" for L/K a finite extension of number fields, generalizing the Pólya group. Using cohomological tools in [1], we compute some relative Pólya groups. As a consequence, we generalize Leriche's results in [17] and prove the triviality of relative Pólya group for the Hilbert class field of K. Then we generalize our previous results [19] on Pólya S 3extensions of Q to dihedral extensions of Q of order 2l, for l an odd prime. We also improve Leriche's upper bound in [16] on the number of ramified primes in Pólya D l -extensions of Q and prove that for a real (resp. imaginary) Pólya D l -extension of Q at most 4 (resp. 2) primes ramify.

A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

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Recently, the data-selective adaptive Volterra filters have been proposed;however, up to now, there are not any theoretical analyses on its behavior rather than numerical simulations. Therefore, in this paper, we analyze the robustness (in the sense of l_2-stability) of the data-selective Volterra normalized least-mean-square (DSVNLMS) algorithm. First, we study the local robustness of this algorithm at any iteration, then we propose a global bound for the error/discrepancy in the coefficient vector. Also, we demonstrate that the DS-VNLMS algorithm improves the parameter estimation for the majority of the iterations that an update is implemented. Moreover, we also prove that if the noise bound is known, then we can set the DS-VNLMS so that it never degrades the estimate. The simulation results corroborate the validity of the executed analysis and demonstrate that the DS-VNLMS algorithm is robust against noise, no matter how its parameters are adopted.

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