Quadratization is a transform of system ODEs with polynomial right-hand side into at most quadratic via the introduction new variables. It has been recently used as pre-processing step for model order reduction methods, so it important to keep number variables small. Several algorithms have designed search quadratization being monomials in original To understand limitations and potential ways i...

let  $r$ be a commutative noetherian ring and $i$ be an ideal of $r$. we say that $i$ satisfies the persistence property if  $mathrm{ass}_r(r/i^k)subseteq mathrm{ass}_r(r/i^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{ass}_r(r/i)$ denotes the set of associated prime ideals of $i$. in this paper, we introduce a class of square-free monomial ideals in the polynomial ring  $r=k[x_1,ld...

The multigraded Poincaré-Betti series P k R(x̄; t) of a monomial ring k[x̄]/〈M〉 on a finite number of monomial generators has the form ∏ xi∈x̄ (1+xit)/bR,k(x̄; t), where bR,k(x̄; t) is a polynomial depending only on the monomial set M and the characteristic of the field k. I present a computer program designed to calculate the polynomial bR,k for a given field characteristic and a given set of monom...

‎‎we investigate graham higman's paper emph{enumerating }$p$emph{-groups}‎, ‎ii‎, ‎in which he formulated his famous porc conjecture‎. ‎we are able to simplify some of the theory‎. ‎in particular‎, ‎higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎emph{monomial} equations is porc‎. ‎it turn...

We study blowups of affine n-space with center an arbitrary monomial ideal and call monomial ideals that render smooth blowups tame ideals. We give a combinatorial criterion to decide whether the blowup is smooth and apply this criterion to discuss a smoothing procedure proposed by Rosenberg, monomial building sets and permutohedra.

d-Monomial tests are statistical randomness tests based on Algebraic Normal Form representation of a Boolean function, and were first introduced by Filiol in 2002. We show that there are strong indications that the Gate Complexity of a Boolean function is related to a bias detectable in a d-Monomial test. We then discuss how to effectively apply d-Monomial tests in chosen-IV attacks against str...

We describe the (co)homology of a certain family of normal subgroups of right-angled Artin groups that contain the commutator subgroup, as modules over the quotient group. We do so in terms of (skew) commutative algebra of squarefree monomial ideals.

We show that certain two-dimensional, integrally closed monomial modules can be uniquely written as a countable product of isomorphic copies of simple integrally closed monomial ideals.