On the Non-Termination of Ruppert’s AlgorithmThis note is an extension of “On the Termination of Ruppert’s Algorithm” which appeared in the Research Notes of the 19th International Roundtable, 2010. While generated after the submission deadline, these results were presented at the conference with the original research note.

Definition 1.2 .

Let $\mathcal{G}:=\{(x,k,y)\in\partial T\times\mathds{Z}\times\partial T:x\thicksim% _{k}y\}$ . For pairs in $\mathcal{G}^{2}:=\{((x,k,y),(y,m,z)):(x,k,y),(y,m,z)\in\mathcal{G}\}$ , we define

(x,k,y)\cdot(y,m,z)=(x,k+m,z).

(1.1)

For arbitrary $(x,k,y)\in\mathcal{G}$ , we define

(x,k,y)^{-1}=(y,-k,x).

(1.2)

Definition 3.5 .

A pair of intervals $([i,j],[i^{{}^{\prime}},j^{{}^{\prime}}])$ is $compatible$ if

[i,j]\cap[i^{{}^{\prime}},j^{{}^{\prime}}]=\emptyset\ \mathrm{or}\ [i,j]% \subset[i^{{}^{\prime}},j^{{}^{\prime}}]\ \mathrm{or}\ [i^{{}^{\prime}},j^{{}^% {\prime}}]\subset[i,j].

Definition 1

Let $\mathcal{B}$ be a $k$ –basis of an algebra $\mathcal{A}$ . $\mathcal{B}$ is a multiplicative basis for $\mathcal{A}$ if

b,b^{\prime}\in\mathcal{B}\Rightarrow b\cdot b^{\prime}\in\mathcal{B}\text{ or% }b\cdot b^{\prime}=0.

Definition 1.2

We say that $r\in R$ is irreducible if

r=ab\ \Rightarrow\ a\ \textrm{is a unit or }\ b\ \textrm{is a unit}.

Definition 2.7 .

Let $f(x)$ be a tropical rational function if there are two tropical polynomials $g(x)$ , $h(x)$ such that

f(x)=h(x)\oslash g(x).

Definition 2.6 .

A multiplicative character on $\mathbb{F}_{q}^{*}$ is a map $\chi$ from $\mathbb{F}_{q}^{*}$ to the nonzero complex numbers $\mathbb{C}^{*}$ that satisfies $\chi(ab)=\chi(a)\chi(b)$ for all $a,b\in\mathbb{F}_{q}^{*}$ . We extend the definition to the whole field $\mathbb{F}_{q}$ by defining

\chi(0)=\left\{\begin{array}[]{ll}1,&\mbox{$\chi=1$};\\ 0,&\mbox{otherwise.}\end{array}\right.

Définition 2.1.3

Let $E$ be a regular (a,b)-module. The dual $E^{*}$ of $E$ is defined as the $\operatorname{\mathbb{C}}[[b]]-$ module $Hom_{\operatorname{\mathbb{C}}[[b]]}(E,E_{0})$ with the $\operatorname{\mathbb{C}}-$ linear map given by

(a.\varphi)(x)=a.\varphi(x)-\varphi(a.x)

where $E_{0}:=\tilde{\mathcal{A}}\big{/}\tilde{\mathcal{A}}.a\simeq\operatorname{% \mathbb{C}}[[b]].e_{0}$ with $a.e_{0}=0$ .

Definition 1.1 (Hom-associative algebra) .

A Hom-associative algebra is a triple $(A,\cdot,\alpha)$ consisting of a vector space $A$ on which $\cdot:A\otimes A\rightarrow A$ and $\alpha:A\rightarrow A$ are linear maps, satisfying

(1.1)

\alpha(x)\cdot(y\cdot z)=(x\cdot y)\cdot\alpha(z).

Definition 1.5 (Hom-Lie algebra) .

A Hom-Lie algebra is a triple $(\mathfrak{g},[\ ,\ ],\alpha)$ consisting of a vector space $\mathfrak{g}$ on which $[\ ,\ ]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ is a bilinear map and $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ a linear map satisfying

(1.3)		$\displaystyle[x,y]=-[y,x],\quad{\text{(skew-symmetry)}}$
(1.4)		$\displaystyle\circlearrowleft_{x,y,z}{[\alpha(x),[y,z]]}=0\quad{\text{(Hom-% Jacobi condition)}}$

for all $x,y,z$ in $\mathfrak{g}$ , where $\circlearrowleft_{x,y,z}$ denotes summation over the cyclic permutation on $x,y,z$ .

Definition 1.9 (Hom-preLie algebras,) .

A left Hom-preLie algebra (resp. right Hom-preLie algebra) is a triple $(A,\cdot,\alpha)$ consisting of a vector space $A$ , a bilinear map $\cdot:A\times A\rightarrow A$ and a homomorphism $\alpha$ satisfying

(1.5)

\alpha(x)\cdot(y,z)-(x\cdot y)\cdot z=\alpha(y)\cdot(x\cdot z)-(y\cdot x)\cdot% \alpha(z),

resp.

(1.6)

\alpha(x)\cdot(y,z)-(x\cdot y)\cdot\alpha(z)=\alpha(x)\cdot(z\cdot y)-(x\cdot z% )\cdot\alpha(y).

Definition 2.3 (Hom-Zinbiel algebra) .

A Hom-Zinbiel algebra is a triple $(A,\circ,\alpha)$ consisting of a vector space $A$ on which $\circ:A\otimes A\rightarrow A$ and $\alpha:A\rightarrow A$ are linear maps satisfying

(2.4)

\displaystyle(x\circ y)\circ\alpha(z)=\alpha(x)\circ(y\circ z)+\alpha(x)\circ(% z\circ y).

for $x,y,z$ in $A$ .