When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints, this task may simply become an undecidable problem [12, 15]. However, studying the integer solutions of linear systems of equations and inequalities is of practical importance in various areas of scientific computing. Two such areas are combinatorial optimization (in particular, integer linear programming) and compiler optimization (in particular, the analysis, transformation, and scheduling of nested loops in computer programs), where a variety of algorithms solve questions related to the points with integer coordinates in a given polyhedron. Another area is at the crossroads of computer algebra and polyhedral geometry, with topics such as toric ideals and Hilbert bases, see [16], as well as the manipulation of Laurent series, see [1].

We use machine learning to approximate the space of steady states in the shuttle model for the Wnt Signaling Pathway. In this ongoing work, we combine these methods with computational algebra to investigate a question by Gross, Harrington, Rosen, and Sturmfels.

We consider the concept of so-called central path in nonlinear optimization that lies at the heart of interior point methods. Using singularity theory and real algebraic geometry, we provide conditions for the existence, convergence, and analyticity of central paths of nonlinear optimization problems. We also prove quantitative bounds on the convergence rate and analyticity of the central path for the special case of semi-definite optimization.

We present a new GeoGebra Discovery command, ShowProof , which provides a certified step-by-step algebraic proof and complexity grade of planar geometry theorems. The output is available in multiple languages and can be exported as plain text to Mathematica or Maple format for further analysis.

Chemical Reaction Networks (CRNs) are a standard formalism used in chemistry and biology to model complex molecular interaction systems. In the perspective of systems biology, they are a central tool to analyze the high-level functions of the cell in terms of their low-level molecular interactions. In the perspective of synthetic biology, they constitute a target programming language to implement in chemistry new functions either in vitro , in artificial vesicles, or in living cells. In this paper, we describe the CRN synthesis tool part of our CRN modeling and analysis software BIOCHAM (Biochemical Abstract Machine). This compiler transforms any elementary (resp. algebraic) real function into a formal finite CRN to compute it (resp. with absolute functional robustness), through a pipeline of symbolic computation steps, among which quadratization optimization plays a key role to restrict to elementary reactions with at most two reactants and a minimum number of molecular species.

Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA. Please send abstracts to the CCA editors <editors_SIGSAM@acm.org> for consideration.

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 5 th Generation (5G) and beyond communication systems. The mmWave radio waves enable high capacity and directive communication, but suffer from many challenges such as rapid channel variation, blockage effects, atmospheric attenuations, etc. The technique initially performs beam alignment procedure, followed by data transfer in the aligned directions between the transmitter and the receiver [1]. 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. [2], beam alignment procedure is optimized by estimating the beam directions using first order polynomial approximations. Recent learning-based SOTA strategies [3] 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. Therefore, our goal is to optimize the beam alignment cost functions e.g. , 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. Specifically, we aim to reduce the beam search space by estimating approximate beam directions using the polynomial solvers. Here, we assume both transmitter (TX) and receiver (RX) to be equipped with uniform linear array (ULA) configuration, each having only one degree of freedom (d.o.f.) with N t and N r antennas, respectively.

Fewnomial Theory [Kho91] has established bounds on the number of connected components (a.k.a. pieces ) of a broad class of real analytic sets as a function of a particular kind of input complexity, e.g., the number of distinct exponent vectors among a generating set for the underlying ideal. Here, we pursue the algorithmic side: We show how to efficiently compute the exact isotopy type of certain (possibly singular) real zero sets, instead of just estimating their number of pieces. While we focus on the circuit case, our results form the foundation for an approach to the general case that we will pursue later.

We present some current improvements, implemented in the software package GeoGebra Discovery, that combine symbolic computation and graphics algorithms to faithfully visualize (semi-)algebraic expressions. Our implementation allows fluid animation of set of (semi-)algebraic sets of dimension 1 in a desktop application or a web browser. We use the Tarski library to create a cylindrical algebraic decomposition of the input, and its plot2d command, which is processed further in GeoGebra Discovery, to provide the user with a familiar look and feel.

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.