Computational algebra is an area of algebra that seeks efficient algorithms to answer fundamental problems concerning basic algebraic objects (groups, rings, fields etc). For questions about generic computer algebra systems, use ...

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The h-vector of a simplicial complex

Let $S$ be a polynomial ring over a field. I want to find an ideal $ I\subseteq S$ such that $(1,2,3,1,1,1)$ is the $h$-vector of $S/I$. We have a relation between $f$-vector and $h$-vector and ...
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Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked ...
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Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated ...
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First order logic in polynomial equations

Have you ever wondered which points on a conic are the intersections of tangent lines of another surface through the origin? More generally, which points on a shape hold some specified relation to all ...
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How to tell if an ideal is absolutely prime

Consider the ideal $I=(ag-ec-1,ah+bg-cf-de)$ of $R=K[a,b,c,d,e,f,g,h]$. Is $I$ prime when $K=\overline{\mathbb{F}}_2$ is the algebraic closure of a field of 2 elements? Can computers answer this ...
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Does this proof that the resultant provides an upper bound for intersection multiplicity look correct?

Let $f,g \in \mathbb{C}[x,y]$ such that $f(0,0)=g(0,0)=0$ and the varieties $V(f)$ and $V(g)$ are both smooth at $(0,0)$ such that the tangent line of $V(f)$ and $V(g)$ is not the $y$ axis. Let ...
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The Command “TzGoGo” in GAP

I am learning GAP and would like to ask one question about a command called "TzGoGo": If $P$ is a finite presentation of a group $G$, then will the eventual result of the command "TzGoGo(P)" be ...
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Urgently Seeking for Help in Calculating the Abelian Invariants of a Group with GAP

I have asked a more complicated question here but I decided to start with an easier version: Let $K_3$ be the group generated by three elements $a$, $b$, $c$ subject to the relation that every simple ...
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Necessary and sufficient conditions for Hensel lifting in the multidimensional case

in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see ...
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An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
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Binary Algorithms

I was wondering. How is it that integrals are done via computers. I can understand basic functions and exponentials but how do you put an integral into binary form so that a computer can do its magic? ...
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minmax of hamming weight in a basis for a vector space

Consider the vector space $V=\mathbb{Z}_2^n$ and take some linear subspace $U\subseteq V$ (we can assume we are given some basis for $U$). Now, for every basis $B$ of U, define $f(B)=\max_{b\in ...
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Find a structure for Hilbert series

Assume $I$ is a monomial ideal in $R=k[x,y]$, $I=\langle m_1,\ldots,m_t\rangle;\ m_i= x^{a_i}y^{b_i}$. I want to find a structure for Hilbert series $R/I$ which depends on $ a_i$ and $b_i$ such that ...
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Exponent in a modular equation

Let $g^i = a \mod p$ and $g^x = a \mod p^2$. Here $p$ is a prime and $g$ generates both $\Bbb Z/{p}\Bbb Z$ and $\Bbb Z/{p^2}\Bbb Z$. Given $i$, $a$ and $p$, how do you find $x$?