Beamforming is a signal processing technique where an array of antenna elements can be steered to transmit and receive radio signals in a specific direction. The usage of millimeter wave (mmWave) frequencies and multiple input multiple output (MIMO) beamforming are considered as the key innovations of 5th Generation (5G) and beyond communication systems. The technique initially performs a beam alignment procedure, followed by data transfer in the aligned directions between the transmitter and the receiver. Traditionally, beam alignment involves periodical and exhaustive beam sweeping at both transmitter and the receiver, which is a slow process causing extra communication overhead with MIMO and massive MIMO radio units. In applications such as beam tracking, angular velocity, beam steering etc., the beam alignment procedure is optimized by estimating the beam directions using first order polynomial approximations. Recent learning-based SOTA strategies for fast mmWave beam alignment also require exploration over exhaustive beam pairs during the training procedure, causing overhead to learning strategies for higher antenna configurations. In this work, we first optimize the beam alignment cost functions e.g. the data rate, to reduce the beam sweeping overhead by applying polynomial approximations of its partial derivatives which can then be solved as a system of polynomial equations using well-known tools from algebraic geometry. At this point, a question arises: 'what is a good polynomial approximation?' In this work, we attempt to obtain a 'good polynomial approximation'. Preliminary experiments indicate that our estimated polynomial approximations attain a so-called sweet-spot in terms of the solver speed and accuracy, when evaluated on test beamforming problems. less

We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized and well structured system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schr\"ocker and \v{S}\'ir 2023). The third method generalizes the dual approach of (Pottmann 1995) from the planar to the spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples. less

We extend the shell and kernel reductions for hyperexponential functions over the field of rational functions to a monomial extension. Both of the reductions are incorporated into one algorithm. As an application, we present an additive decomposition in rationally hyperexponential towers. The decomposition yields an alternative algorithm for computing elementary integrals over such towers. The alternative can find some elementary integrals that are unevaluated by the integrators in the latest versions of Maple and Mathematica. less

In this paper we introduce a novel quantifier elimination method for conjunctions of linear real arithmetic constraints. Our algorithm is based on the Fourier-Motzkin variable elimination procedure, but by case splitting we are able to reduce the worst-case complexity from doubly to singly exponential. The adaption of the procedure for SMT solving has strong correspondence to the simplex algorithm, therefore we name it FMplex. Besides the theoretical foundations, we provide an experimental evaluation in the context of SMT solving. less