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### Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let $k$ be a field and let $R=k[X_1,...,X_n]$ be a polynomial ring. Let $F \subset R$ be a finite subset generating a radical ideal $I$ with precisely $e$ solutions over an algebraic closure of $k$. ...
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### Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots, x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl algebra. As usual $W$ ...
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### Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n]$ be the nth Weyl algebra over the characteristic zero field $k$. Set $\theta_i=x_i\partial_i$. Let $I$ be a left ideal in $D$. Is there a ...
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### Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...
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### Computations in Weyl algebra with rational function coefficients

I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational ...
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### Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
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### Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...
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### How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation $qn = mf$, where each of the variables ...
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### Polynomial constraints triggered by irreducibility [closed]

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic: $$af^2 + bf + c = 0$$ If we're working in a ring, ...
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### Fast computation of a Groebner basis - What is Possible

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
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### Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$. In what follows, the height of an algebraic number will mean the absolute multiplicative height. Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
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### Degenerations and spanning monomials

Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...
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### Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer. Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
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### Dimension of a homogeneous polynomial system

Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations: \begin{cases} &\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}...
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### Algorithm to detect if an element of a (commutative) ring is in a subring?

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring. Is there ...
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### How to recover the ideal from grobner basis of kernel of ann(x)

M -> ann(x) i can find the grobner basis of kernel of ann(x) and need the final step to recover this basis to ideal as i know, eliminate is not for all cases, what is the general practice to treat ...
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### Can we find a Groebner Basis?

I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We ...
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### Groebner bases for power series rings (reference request)

Hello, Could you help me with a reference to elementary properties of Groebner bases in rings of formal power series over a field? I am especially interested in generic initial ideals. Thank you in ...
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### Solving the Field Membership Problem using Grobner Bases

Is there an easy way to determine whether a set of elements in a field generates the whole field or only a subfield? Specifically, I have a subfield of $k(x,y)$ described in terms of a canonical set ...
Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and $C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated $\... 1answer 421 views ### Reasonable implementation of finding Gröbner bases over non-field coefficient rings Gröbner bases are usually considered in the ring of polynomials over a field. However, there are useful definitions and algorithms for Gröbner bases over other coefficient rings; see, for instance, ... 2answers 326 views ### Monomial orderings in noncommutative setting An ordering of monomials in the free associative algebra$k\langle x_1,\ldots,x_n\rangle$is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ... 0answers 246 views ### Bounding the degrees in a Bézout relation for integer polynomials Let$A$and$B$be two polynomials in$\mathbf Z[X]$which generate$\mathbf Z[X]$, that is assume that there exist polynomials$U$and$V$in$\mathbf Z[X]$such that $$A \cdot U + B \cdot V=1.$$ ... 1answer 240 views ### What does the d-slice of a weighted polynomial algebra look like? This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves. ... 1answer 404 views ### Can we make Buchberger's algorithm faster for a given ideal if we are allowed to vary the monomial order? Suppose we have a finite set of generators for an ideal$I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where$\Bbbk$is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to ... 1answer 1k views ### Existence of a real-valued solution to system of multivariate polynomial equations Given a system of multivariate, polynomial equations, is there a way to determine if it has a solution in a given field (for instance the set of all reals). I don't care what the solution is, I just ... 1answer 619 views ### PBW theorem over a Q-algebra, without freeness or flatness Let$k$be a commutative ring with$1$. Let$L$be a$k$-Lie algebra, which is not necessarily free as a$k$-module. Let$S\left(L\right)$denote the symmetric algebra of$L$(over$k$), constructed ... 2answers 471 views ### Nonstandard monomial orders? Are there any articles/books/examples where a non-standard monomial order is used? What are the applications of these monomial orders? In particular, uses in groebner bases and variable elimination. (... 2answers 2k views ### Numerical solution for a system of multivariate polynomial equations Hi all, I have a system of 6th-order polynomial equations in 4 variables$q_1, q_2, q_3, q_4$(i.e. polynomials with all the terms such as$q_1^6, q_2^6, q_2^4 q_3^2$):$P_k(q_1, q_2, q_3, q_4) = 0$... 1answer 350 views ### Is the first part of Eisenbud's Proposition 15.15's proof o.k? In the chapter on Gröbner bases from Eisenbud's "Commutative Algebra" the following statement appears as Proposition 15.15 (page 344): Let$F$be a free$S$module with basis and monomial order ... 4answers 985 views ### Systems of polynomial equations Hi all, I'm an engineer assigned to determine some parameters of a manipulator (ie., calibration). It has a number of parameters, but after some manipulations of its dynamic equations, I can have the ... 2answers 632 views ### Differential ideal membership problem We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case? To be ... 4answers 361 views ### Lower bounds on the degrees of representatives of$u^n$as$n \to \infty$Let$k$be an algebraically closed field and$A$a finitely generated$k$-algebra, together with a specified surjective morphism$\phi \colon k[x_1, \dotsc, x_n] \to A$. For$f \in A$, define$\...
Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...