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A question on bounding the size of the polynomial

Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$: $f_1 = x_1 + x_n^2$ $f_2 = x_2 + x_1^2$ $\cdot$ $\cdot$ $f_{n-3} = x_{n-3} + x_{n-4}^2$ $f_{n-2} = x_{n-2} + x_{n-...
Rishabh Kothary's user avatar
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42 views

When $x=\frac{u(f_i,g_j)}{v(f_i,g_j)}$ implies $x=\frac{u(f_i(x,0),g_j(x,0))}{v(f_i(x,0),g_j(x,0))}$ ($x=\frac{xy}{y}$ does not imply $x=\frac{0}{0}$)

Let $f_i=f_i(x,y), g_j=g_j(x,y) \in \mathbb{C}[x,y]$, $1 \leq i \leq n$, $1 \leq j \leq m$, be such that $f_i(x,0) \neq 0$ and $g_j(x,0)=0$. Assume that $\mathbb{C}(f_1,\ldots,f_n,g_1,\ldots,g_m)=\...
user237522's user avatar
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31 views

Formalization of the independance of products in a (commutative) semigroup

1/ It is well known that associativity implies that the result of the product of an ordered finite set of elements in a semigroup does not depend of the order of composition of the partial products. ...
Gérard Lang's user avatar
  • 2,655
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47 views

Relationship between equation of integral dependence of an element and its inverse

Let $A$ be a reduced, Noetherian ring. Let $B$ be its integral closure. Let $b\in B$ and let $v\in B$ be its inverse. Let $b^n+\ldots a_0=0$ be an equation of integral dependence for $b$. Is there any ...
Rudyard's user avatar
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97 views

Algebraic independence and substitution for quadratics

Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
Rishabh Kothary's user avatar
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76 views

Largest set of monomials whose span is "co-prime" to a given polynomial

Let $K$ be a number field, and let $F \in K[x_1, \cdots, x_n]$ be a polynomial. For a positive integer $d \geq 3$, define $M(F;d)$ to be the largest positive integer such that there exists a set $S$ ...
Stanley Yao Xiao's user avatar
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113 views

Relation between minimality and algebraic independence for binomials?

$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that $f_1 = x_1 + q_1$ $f_2 = x_2 + q_2$ $\cdot \cdot \cdot$ $f_{n-1} = x_{n-1} + q_{n-1}$ $f_{n} = q_n$ such that ...
Rishabh Kothary's user avatar
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82 views

Integer valued polynomials and divided power algebra

Let $T\subset \mathbb Q[x]$ be the ring of integer valued polynomials, i.e. the polynomials $f$ with $f(\mathbb Z)\subset \mathbb Z$. In his wonderful book ”Commutative algebra with a view toward ...
Kasper Andersen's user avatar
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95 views

Which algebraic structure characterizes the set of non-trivial qudratic residues in a finite field?

I understand this question may be too naive to ask, but I am unable to figure it out. Suppose, $\mathbb{QR^*}$ denotes the set of all quadratic residues in a finite field except the identity element $...
Somudro Gupto's user avatar
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57 views

Reference for packing property and König property

Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
Sowbarnika R's user avatar
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100 views

Shedding faces and decomposability in simplicial complexes

Definition: A pure d-dimensional complex $\Delta$ is $k$-decomposable if either $\Delta$ is a $d-$simplex or $\Delta$ contains a face $F$ such that $\dim(F) \leq k$ both $\Delta \setminus F$ and $\...
user177523's user avatar
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87 views

Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?

Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
José's user avatar
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184 views

Degree 6 Galois extension over $\mathbb{Q} $

Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
Sky's user avatar
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70 views

Hensel lifting of roots of a biquadratic polynomial

Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
HIMANSHU's user avatar
  • 381
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60 views

Symbolic polyhedron of a monomial ideal

$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then $$ \alpha(P)= \min \{...
Sowbarnika R's user avatar
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111 views

Totally isotropic space for bilinear pairing over ring

A duplicate of this: Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$ a totally ...
JBuck's user avatar
  • 223
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109 views

Affine scheme over ring of meromorphic functions with finite poles on unit circle

I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
Jens Fischer's user avatar
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329 views

Smooth morphisms under base change, Qing Liu's proposition 4.3.38

I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
BernyPiffaro's user avatar
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126 views

Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$

This is generalization of the univariate case and also related to open problem. Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
joro's user avatar
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117 views

A question on a system of quadratic polynomials

Consider the following system of quadratic polynomials $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,....,x_n]$ : $f_1 (\bar{x}) = x_1 + x_n^2 + q_1$ $f_i(\bar{x}) = x_i + q_i$ for $i \in \{2,...,n-1 \}$ $...
Rishabh Kothary's user avatar
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71 views

"Approximating" ring of semi-invariants

I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
It'sMe's user avatar
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163 views

A complex with homology $=R/p$

Given a Noetherian ring $R$ . I am looking for a bounded complex $X$ of finitel geenerated projectives over $R$ whose homology is $R/p$. Infact I just need $X$ to have $\operatorname{Supp}(H(X)) = \...
Subham Jaiswal's user avatar
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92 views

What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?

What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known? Given that there are $3{,}684{,}030{,}417$ semigroups of order $8$, I guess $n\...
Shaun's user avatar
  • 379
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0 answers
78 views

Let $A, B$ be matrices with elements in $\mathbb{Z}_n$, does $\ker A = \ker B$ imply that they are row equivalent?

Let $A, B$ be matrices with elements in $\mathbb{Z}_n$. If $A x = 0$ and $B x = 0$ have the same set of solutions, where the vectors also have elements in $\mathbb{Z}_n$, does this mean that there is ...
José's user avatar
  • 219
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0 answers
90 views

Invariants of primary groups

In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...
Nini's user avatar
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0 answers
177 views

Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced

I have a question about correctness of following statement claimed here in $\boxed{2} \ $: Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$-...
user267839's user avatar
  • 6,038
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0 answers
124 views

Krull dimension of ring of invariants

Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
kindasorta's user avatar
  • 2,907
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0 answers
213 views

Taking polynomial inverses over a field?

Let $f \in F_p[x] / q(x)$. I know for a fact that the inverse of $f$, $g$ exists in the field. Taking the regular inverse is easy, but I'm looking for the compositional inverse. I'm looking for ...
mtheorylord's user avatar
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0 answers
63 views

A construction that sort of merges two semigroups to build a new one

Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
Salvo Tringali's user avatar
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122 views

Is there a name for this condition on a monoid?

Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
Steven Stadnicki's user avatar
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156 views

Absolute integral closure of Noetherian local domain

Let $(R,\mathfrak{m})$ be a Noetherian local domain, let $K(R)$ be the total fraction field of $R$ and let $R^{+}$ be the absolute integral closure of $R$. Consider the $R^{+}$-module $K(R)\otimes_{R} ...
CARLO's user avatar
  • 39
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0 answers
215 views

On linear schemes

Let $X$ be a smooth projective curve and $T$ is any smooth variety. Let $E$ be a family of vector bundles over $X\times T$ which is flat over $T$. Then there exists a scheme $Y$ over $T$ such that for ...
S.D.'s user avatar
  • 494
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0 answers
179 views

Product/intersection of two ideals

Let $I_{1}$ and $I_{2}$ be two ideals in polynomial ring $C[u,v,t]$, defined as $I_1 = \langle t-u^3, (v-u^5)^2\rangle$ and $I_2 = \langle u^{11} + v^{11} + t^{11} + 3\rangle$. Is there a method for ...
Ethan's user avatar
  • 1
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0 answers
91 views

Comparison of depth of two monomial ideals

Let $I$ and $J$ be two monomial ideals of $R=K[x_1,\dotsc,x_n]$ where $K$ is a field. Could we say $\mid\operatorname{depth} (R/I)- \operatorname{depth}(R/(I:J))\mid\leq 1$, where $(I:J)$ is colon ...
Amir Mafi's user avatar
  • 113
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0 answers
61 views

A sequence of polynomials that the variety defined by every $n$ of them is small

Let $\mathbb{C}[x_1,x_2,\cdots,x_n]_{= d}$ denote the set of polynomials in $\mathbb{C}[x_1,x_2,\cdots,x_n]$ of total degree $d$. Is there exists a sequence of polynomials $f_1,f_2,f_3,\cdots$ in $\...
Yuting Li's user avatar
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0 answers
28 views

Differentials for a nonsigular subvarity of a nonsigular variety

Let $A$ be a noetherian regular local ring of dimension $n$, and $P\subset A$ a prime ideal, such that $A/P$ still a regular local ring of dimension $m$. I want to show $P/P^2$ is a $n-m$ rank free ...
ZhouQi's user avatar
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0 answers
226 views

Geometric interpretation of normalization inside a finite extension of function field

$\DeclareMathOperator\Spec{Spec}$Suppose $X = \Spec A$ is a smooth affine variety over $\mathbb C$ and suppose $L/K$ is a finite extension of its function field. Let $Y = \Spec B$, where $B$ is the ...
Mohan Swaminathan's user avatar
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53 views

When a given set of primes of height 1 is a set associated primes of an element

Let $R$ be a Noetherian local ring of dimension $\geq 3$ and $\{p_1,\ldots , p_n\}$ be a collection of prime ideals of height $1$. Does there exist an element $x\in R$ such that $Ass(R/xR)=\{p_1,\...
Cusp's user avatar
  • 1,713
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0 answers
172 views

When does this commutative non-associative algebra have nilpotent elements?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dotsc, ...
mick's user avatar
  • 763
0 votes
1 answer
362 views

Derivations and ideals

Let $R$ be a regular local ring and $I$ and ideal of $R$. If $D$ is a derivation of $R$, let $$\lambda_D:I/I^2\to R/I$$ be the composition of the restriction of $D$ to $I$ and the quotient map $R\to R/...
Hephaistos's user avatar
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0 answers
267 views

completion and tensor product

Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers? If $A$ is noetherian, it is clear because one has for $k$ a residue ...
prochet's user avatar
  • 3,472
0 votes
0 answers
133 views

Example of a periodic free resolution over a hypersurface

I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION, WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud I'm wondering what would be a nice example illustrating Theorem 6....
It'sMe's user avatar
  • 839
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0 answers
130 views

Is a closed subsecheme contained in a Cartier divisor?

Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...
OOOOOO's user avatar
  • 349
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0 answers
89 views

Representing an $m$ dimensional quadratic polynomial as a polynomial on $\mathbb F_{q^m}$

We can represent $\mathbb{F}_{q^m}$ as $\mathbb{F}_q[\alpha]$ where $\alpha$ is root of an irreducible $m$-degree polynomial on $\mathbb{F}_q$. By sending $\sum_{i=0}^{m-1} c_i\alpha^i \mapsto (c_{m-1}...
user's user avatar
  • 1
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0 answers
130 views

Complex dimension of zeros of vanishing ideal vs real dimension

Let $S \subseteq \mathbb{R}^n$ be a subset of real points and $I(S)$ be the vanishing ideal of $S$ in $\mathbb{R}[x_1,\dotsc,x_n]$. Is $\dim V_{\mathbb{R}}(I(S)) = \dim V_{\mathbb{C}}(I(S))$? I.e., is ...
Pew's user avatar
  • 263
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0 answers
350 views

Relation between $3$-term Plücker relations and more than $3$-term Plücker relations

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...
Jianrong Li's user avatar
  • 6,201
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0 answers
41 views

Polyextremal groups

A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form $f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
Taras Banakh's user avatar
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0 answers
62 views

To find a DFT for complex functions on a semigroup

For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
CommonAnts's user avatar
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0 answers
132 views

Which consequences can be deduced from this peculiar property of tetration?

Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...
Marco Ripà's user avatar
  • 1,451
0 votes
0 answers
162 views

Finite-exponent abelian groups

Let $G$ be an abelian group and $G=\bigoplus_{i=1}^t{{\Bbb{Z}}_{p_i}^{n_i}}^{(\Lambda_i)}$ where each $\Lambda_i$ is a set (at least one of $\Lambda_i$ is infinite). Since $G_{\Bbb{Z}}$ is a finite-...
Najmeh Dehghani's user avatar

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