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Additive regression models have a long history in multivariate nonparametric regression. They provide a model in which each regression function depends only on a single explanatory variable allowing to obtain estimators at the optimal…
The theme of symbolic computation in algebraic categories has become of utmost importance in the last decade since it enables the automatic modeling of modern algebra theories. On this theoretical background, the present paper reveals the…
Polynomial filters, a kind of Graph Neural Networks, typically use a predetermined polynomial basis and learn the coefficients from the training data. It has been observed that the effectiveness of the model is highly dependent on the…
In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the…
This paper presents an algorithm for computing Groebner bases based upon labeled polynomials and ideas from the algorithm F5. The main highlights of this algorithm compared with analogues are simplicity both of the algorithm and of the its…
We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials…
Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded as a set of polynomials that automatically generates a Gr\"obner basis…
Mixtures of Mallows models are a popular generative model for ranking data coming from a heterogeneous population. They have a variety of applications including social choice, recommendation systems and natural language processing. Here we…
In 1965 Buchberger defined Gr\"obner bases and an algorithm to compute them. Despite a slow start, already in the eighties Gr\"obner bases had become the main device for symbolic computations involving polynomials as well as a theoretical…
In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two…
Given an affine algebra $R=K[x_1,\dots,x_n]/I$ over a field $K$, where $I$ is an ideal in the polynomial ring $P=K[x_1,\dots,x_n]$, we examine the task of effectively calculating re-embeddings of $I$, i.e., of presentations $R=P'/I'$ such…
We report on our experiences exploring state of the art Groebner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature…
Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple classifiers. This chapter provides an analytical…
Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…
In this paper we present the first-ever computer formalization of the theory of Gr\"obner bases in reduction rings, which is an important theory in computational commutative algebra, in Theorema. Not only the formalization, but also the…
This short note is the generalization of Faugere F4-algorithm for polynomial rings with coefficients in Euclidean rings. This algorithm computes successively a Groebner basis replacing the reduction of one single s-polynomial in…
Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enable…
Signature-based algorithms have become a standard approach for Gr\"obner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this…
Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of…
The Gr\"obner basis detection (GBD) is defined as follows: Given a set of polynomials, decide whether there exists -and if "yes" find- a term order such that the set of polynomials is a Gr\"obner basis. This problem was shown to be NP-hard…