Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gröbner and Border bases are fundamental but suffer from high computational costs, which have motivated recent Deep Learning approaches to improve efficiency, albeit at the expense of output correctness. In this work, we introduce the Oracle Border Basis Algorithm, the first Deep Learning approach that accelerates Border basis computation while maintaining output guarantees. To this end, we design and train a Transformer-based oracle that identifies and eliminates computationally expensive reduction steps, which we find to dominate the algorithm's runtime. By selectively invoking this oracle during critical phases of computation, we achieve substantial speedup factors of up to 3.5x compared to the base algorithm, without compromising the correctness of results. To generate the training data, we develop a sampling method and provide the first sampling theorem for border bases. We construct a tokenization and embedding scheme tailored to monomial-centered algebraic computations, resulting in a compact and expressive input representation, which reduces the number of tokens to encode an $n$-variate polynomial by a factor of $O(n)$. Our learning approach is data efficient, stable, and a practical enhancement to traditional computer algebra algorithms and symbolic computation.

Solving a polynomial system, or computing an associated Gröbner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the worst case. This paper is the first to address the learning of Gröbner basis computation with Transformers. The training requires many pairs of a polynomial system and the associated Gröbner basis, raising two novel algebraic problems: random generation of Gröbner bases and transforming them into non-Gröbner ones, termed as backward Gröbner problem. We resolve these problems with 0-dimensional radical ideals, the ideals appearing in various applications. Further, we propose a hybrid input embedding to handle coefficient tokens with continuity bias and avoid the growth of the vocabulary set. The experiments show that our dataset generation method is a few orders of magnitude faster than a naive approach, overcoming a crucial challenge in learning to compute Gröbner bases, and Gröbner computation is learnable in a particular class.

In Commutative Algebra and Algebraic Geometry, ''Primary decomposition'' is well-known as a fundamental and important tool. Although algorithms for primary decomposition have been studied by many researchers, the development of fast algorithms still remains a challenging problem. In this paper, we devise an algorithm for ''Strong Intermediate Primary Decomposition" via maximal independent sets by using modular techniques. In the algorithm, we utilize double ideal quotients to check whether a candidate from modular computations is an intersection of prime divisors or not. As an application, we can compute the set of associated prime divisors from the strong intermediate prime decomposition. In a naive computational experiment, we see the effectiveness of our methods.

By double ideal quotient, we mean (I : (I : J)) where I and J are ideals. In our previous work [12], double ideal quotient and its variants are shown to be very useful for checking prime divisors and generating primary components. Combining those properties, we can compute "direct localization" effectively, comparing with full primary decomposition. In this paper, we apply modular techniques effectively to computation of such double ideal quotient and its variants, where first we compute them modulo several prime numbers and then lift them up over rational numbers by Chinese Remainder Theorem and rational reconstruction. As a new modular technique for double ideal quotient and its variants, we devise criteria for output from modular computations. Also, we apply modular techniques to intermediate primary decomposition. We examine the effectiveness of our modular techniques for several examples by preliminary computational experiments in Singular.

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gröbner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gröbner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting.

This paper presents the integrated circuit (IC) which could output a driving waveform to generate the walking motion of the piezoelectric element impact-type micro electro mechanical systems (MEMS) microrobot. The microrobot was made from silicon wafer fabricated by micro fabrication technology. The size of the fabricated robot was 4.0 × 4.6 × 3.6 mm. IC design of the pulse-type hardware neuron model (P-HNM) had been done by using CMOS process. P-HNM has the same basic features of biological neurons to generate the pulse waveform. Therefore, P-HNM outputs the driving waveform using electrical oscillation such as biological neuron. In this paper, we showed that the P-HNM which outputs the driving waveform for the piezoelectric element impact-type MEMS actuator could design as bare chip IC. As a result, we showed that P-HNM with driving circuit could generate the driving waveform of the rotary-type actuator of piezoelectric element impact-type MEMS microrobot. The generation of the driving waveform could realize without any software programs or analog digital converters.