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In this paper, we present an algorithm for determining the limits of real rational functions in two variables, based on Sturm's familiar theorem and the general Sturm–Tarski theorem for counting certain roots of univariate polynomials in a real closed field. Let R [ x , y ] be the ring of polynomials with real coefficients in two variables x , y , and let u (x , y) , v (x , y) ∈ R [ x , y ] be two non-zero polynomials such that u (a , b) = v (a , b) = 0 for a , b ∈ R. The purpose of this paper is to decide the existence of lim (x , y) → (a , b) ⁡ u (x , y) v (x , y) and compute the limit if it exists. Our algorithm needs no assumption on the denominators and does not involve the computation of Puiseux series. [ABSTRACT FROM AUTHOR]

Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f ∈ K [ X ] = : R and computes the invariant ring (R f) G together with a presentation. The morphism X f → Spec ((R f) G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gröbner basis computations. If R is a complete intersection, then so is (R f) G. [ABSTRACT FROM AUTHOR]

Powers of (monomial) ideals is a subject that still calls attraction in various ways. In this paper we present a nice presentation of high powers of ideals in a certain class in K [ x 1 , ... , x n ] and K [ [ x 1 , ... , x n ] ]. As an interesting application it leads to an algorithm for computation of the Ratliff–Rush operation on ideals in that class. The Ratliff–Rush operation itself has several applications, for instance, if I is a regular m -primary ideal in a local ring (R , m) , then the Ratliff–Rush associated ideal I ˜ is the unique largest ideal containing I with the same Hilbert polynomial as I. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of special ideals. We generalize the results on those as well as the algorithm obtained for finite dimensional power series rings by Mordechai Katzman and Wenliang Zhang to finite dimensional polynomial rings. [ABSTRACT FROM AUTHOR]

The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The result does not extend directly to polynomials over semirings, but we do have analogous results for some special semirings, for example, the tropical, extended and supertropical semirings. These all fall into a larger class of upper-bound semirings. In this paper we extend the known results and give a complete characterization of elementary upper-bound semirings. We further improve this characterization statement in the case of linearly ordered upper-bound semirings. [ABSTRACT FROM AUTHOR]

In this study, we determine the structure of cyclic codes over the ring Z q [ u ] / 〈 u 2 〉 which is isomorphic to R = Z q + u Z q where q = p s , p is a prime, s is a positive integer, and u 2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R [ x ] / 〈 x n − 1 〉 , which is addressed in this paper completely. By establishing the structure of ideals of R [ x ] / 〈 x n − 1 〉 with gcd ⁡ (p , n) = 1 , we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R / I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R / I. This is exploited in this paper to introduce the use of two special (non-monomial) types of basis functions with nice properties. This allows us, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study the bound of the number of limit cycles by Poincaré bifurcation for a Liénard system of type ( 4 , 3 ) . An automatic algorithm is constructed based on the Chebyshev criteria and the tools of regular chain theory in polynomial algebra. We prove the system can bifurcate at most 6 limit cycles from the periodic annulus by this algorithm and at least 4 limit cycles by asymptotic expansions of the related Melnikov functions. [ABSTRACT FROM AUTHOR]

In this paper we study standard bases for submodules of a mixed power series and polynomial ring R 〚 t 1 , … , t m 〛 [ x 1 , … , x n ] s respectively of their localisation with respect to a t _ -local monomial ordering for a certain class of noetherian rings R , also called Zacharias rings. The main steps are to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora and to generalise the Buchberger criterion. Everything else then translates naturally. Setting either m = 0 or n = 0 we get standard bases for polynomial rings respectively for power series rings over R as a special case. [ABSTRACT FROM AUTHOR]

Let R be a non-commutative PID finitely generated as a module over its center C . In this paper we give a criterion to decide effectively whether two given elements f , g ∈ R are similar, that is, if there exists an isomorphism of left R -modules between R / R f and R / R g . Since these modules are of finite length, we also consider the more general problem of deciding when two given left R -modules of finite length are isomorphic. This criterion allows the design of algorithms when R is an Ore extension of a skew-field whose center is a commutative polynomial ring. We propose two methods which, essentially, check the equality of the rational canonical forms of certain matrices with coefficients in C associated to each of the modules. These algorithms are based on the fact that, if R is finitely generated as a C -module, then the existence of an isomorphism of R -modules can be reduced to checking the existence of an isomorphism of C -modules. Actually, we prove this result in the realm of non-commutative principal ideal domains, generalizing a version given by Jacobson for some Ore extensions of a skew field by an automorphism. [ABSTRACT FROM AUTHOR]

We call an ideal in a polynomial ring robust if it can be minimally generated by a universal Gröbner basis. In this paper we show that robust toric ideals generated by quadrics are essentially determinantal. We then discuss two possible generalizations to higher degree, providing a tight classification for determinantal ideals, and a counterexample to a natural extension for Lawrence ideals. We close with a discussion of robustness of higher Betti numbers. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend the characterization of , where to be a free -module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a -vector space basis of residue class polynomial rings over a field (Macaulay–Buchberger Basis Theorem) to rings, i.e. , where is an ideal. We give some insights into the characterization for two special cases, when and . As an application of this characterization, we show that the concept of Border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free A-module. [Copyright &y& Elsevier]