arXiv subjects

Sheng-Ming Ma

Publications and source records attributed to Sheng-Ming Ma.

5 recordsLinked to original sources

arXiv2026

Beyond F5 and GVW: The Proper-Cover Algorithm for Fast Ideal Basis Computation

Gr\"obner basis computation incurs heavy computational overhead, especially under lexicographic order. F5 and its GVW variant dominate efficient field-based Gr\"obner basis solving. The proper basis algorithm offers a parameterized ideal computation framework without leveraging modern signature-based optimizations. This work presents the Proper-Cover algorithm for zero-dimensional polynomial ideals by combining GVW's cover optimization over signature with the proper basis theory. We generalize signature, cover, POT ordering, reduction and S-pair concepts to parameterized coefficients, design a two-phase algorithm with compatible factor construction and hungry refinement, and rigorously prove termination and output correctness. Accordingly, we propose a new framework for the efficient computation of polynomial ideal bases. Benchmark results show that Proper-Cover surpasses F5 under all monomial orderings and delivers clear speedups over GVW for lexicographic (plex) order.

cs.SC↗

arXiv2022

A New Type of Gröbner Basis and Its Complexity

The new type of ideal basis introduced herein constitutes a compromise between the Gröbner bases based on the Buchberger's algorithm and the characteristic sets based on the Wu's method. It reduces the complexity of the traditional Gröbner bases and subdues the notorious intermediate expression swell problem and intermediate coefficient swell problem to a substantial extent. The computation of an $S$-polynomial for the new bases requires at most $O(m\ln^2m\ln\ln m)$ word operations whereas $O(m^6\ln^2m)$ word operations are requisite in the Buchberger's algorithm. Here $m$ denotes the upper bound for the numbers of terms both in the leading coefficients and for the rest of the polynomials. The new bases are for zero-dimensional polynomial ideals and based on univariate pseudo-divisions. However in contrast to the pseudo-divisions in the Wu's method for the characteristic sets, the new bases retain the algebraic information of the original ideal and in particular, solve the ideal membership problem. In order to determine the authentic factors of the eliminant, we analyze the multipliers of the pseudo-divisions and develop an algorithm over principal quotient rings with zero divisors.

cs.SC↗

arXiv2021

The Proper Basis for Polynomial Ideals

We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The Gr\"obner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in $x_1$. The proper basis is defined and computed in the variables $\tilde{\bm{x}}:=(x_2,\dotsc,x_n)$ with $x_1$ serving as a parameter in the algebra $K[x_1][\tilde{\bm{x}}]$. Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of $\tilde{\bm{x}}$ unnecessarily involves the least variable $x_1$ but also M\"oller's algorithm due to its polynomial division mechanism. This is corroborated by a series of benchmark testings herein. The proper basis is in a modular form and neater than Gr\"obner basis and hence reduces its coefficient swell problem. It is expected that all the state of the art algorithms improving Buchberger's algorithm over the last decades can be further improved if we apply them to the proper basis.

math.AC↗

arXiv2014

A Newtonian and Weierstrassian Approach to Local Resolution of Singularities in Characteristic Zero

This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial transformations and based on Weierstrass preparation theorem. This approach entails such new methods as canonical reduction and synthesis of monomial transformations as well as latency and revival of primary variables. The orders of primary variables serve as the decreasing singularity invariants for the algorithm albeit with some temporary increases. A finite partition of unity in a neighborhood of the singular point is constructed in an inductive way depending on the topological constraint imposed by Euler characteristic of the normal vector set of Newton polyhedron.

math.AG↗

arXiv2010

The sharp bound for the number of real solutions to polynomial equation systems

This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed system of $n$ polynomial equations in $n$ variables, this paper shows that the number of its positive solutions in $\mathbf{R}_*^n$ is sharply bounded by that of the simplex configurations in the triangulation of its support generically. The proof is based on a homotopic argument and an inductive triangulation of the support of the system via a hierarchy of pyramid configurations of different orders. For the mixed system of $n$ polynomial equations in $n$ variables, this paper shows that the maximal number of positive solutions in $\mathbf{R}_*^n$ to the systems with the same support is a symmetric multilinear function of the support generically and hence can be computed via the polarization identity.

math.AG↗