$\big{(}B/xB\big{)}^{(\ell)}$ have the same underlying ring, and since (by
definition) rigidity only depends on the underlying ring, we have shown that

(20) $\text{if $R=B_{(x)}$ is non-rigid then $\big{(}B/xB\big{)}^{(\ell)}$ is
non-rigid}.$

Since (by Lemma 7.19(a)) $\ell=\operatorname{\text{\rm
lcm}}(d,e(B/xB))=e\big{(}(B/xB)^{(d)}\big{)}$, we have
$(B/xB)^{(d)}=\big{(}(B/xB)^{(d)}\big{)}^{(\ell)}$; since $d\mid\ell$, we have
$\big{(}(B/xB)^{(d)}\big{)}^{(\ell)}=(B/xB)^{(\ell)}$; so
$(B/xB)^{(d)}=(B/xB)^{(\ell)}$. This together with (20) completes the proof. ∎

The next fact is an application of the above Theorem. Refer to 8.1 and 8.2 for
the notation.

######  9.8 Proposition.

Let $n\geq 4$ and
${\mathbf{a}}=(a_{1},\dots,a_{n})\in(\mathbb{N}\setminus\\{0\\})^{n}$. Let
${\rm\bf k}$ be a field of characteristic $0$ and consider the Pham-Brieskorn
ring $B_{{\rm\bf k};{\mathbf{a}}}=B_{{\rm\bf k};a_{1},\dots,a_{n}}={\rm\bf
k}[x_{1},\dots,x_{n}]$. For each $i\in\\{1,\dots,n\\}$, let
${\mathbf{a}}(i)=(a_{1},\dots,\widehat{a_{i}},\dots,a_{n})\in(\mathbb{N}\setminus\\{0\\})^{n-1}$
and consider the corresponding ring $B_{{\rm\bf k};{\mathbf{a}}(i)}=B_{{\rm\bf
k};a_{1},\dots,\widehat{a_{i}},\dots,a_{n}}$. Then the implication

(21) $\text{$B_{{\rm\bf k};{\mathbf{a}}(i)}$ is
rigid}\implies\text{$(B_{{\rm\bf k};{\mathbf{a}}})_{(x_{i})}$ is rigid}$

is valid for each $i\in\\{1,\dots,n\\}$ that satisfies
$S({\mathbf{a}})\subseteq\\{i\\}$.

######  Remark.

By definition, $\operatorname{\text{\rm
cotype}}({\mathbf{a}})=|S({\mathbf{a}})|$. Clearly, if
$\operatorname{\text{\rm cotype}}({\mathbf{a}})=0$ then all
$i\in\\{1,\dots,n\\}$ satisfy $S({\mathbf{a}})\subseteq\\{i\\}$; if
$\operatorname{\text{\rm cotype}}({\mathbf{a}})=1$ then exactly one
$i\in\\{1,\dots,n\\}$ satisfies $S({\mathbf{a}})\subseteq\\{i\\}$; and if
$\operatorname{\text{\rm cotype}}({\mathbf{a}})>1$ then no element $i$ of
$\\{1,\dots,n\\}$ satisfies $S({\mathbf{a}})\subseteq\\{i\\}$.

###### Proof of Prop. 9.8.

We may assume that $i=n$, so it suffices to prove that

if $S({\mathbf{a}})\subseteq\\{n\\}$ and $(B_{{\rm\bf
k};{\mathbf{a}}})_{(x_{n})}$ is non-rigid, then $B_{{\rm\bf
k};{\mathbf{a}}(n)}$ is non-rigid.

Suppose that $S({\mathbf{a}})\subseteq\\{n\\}$ and that $(B_{{\rm\bf
k};{\mathbf{a}}})_{(x_{n})}$ is non-rigid; we shall prove that $B_{{\rm\bf
k};{\mathbf{a}}(n)}$ is non-rigid. Thm 9.6 implies that $(B_{{\rm\bf
k};{\mathbf{a}}}/x_{n}B_{{\rm\bf k};{\mathbf{a}}})^{(d_{n})}$ is non-rigid.
Note that $B_{{\rm\bf k};{\mathbf{a}}}/x_{n}B_{{\rm\bf
k};{\mathbf{a}}}\overset{\text{\tiny$*$}}{\cong}B_{{\rm\bf
k};{\mathbf{a}}(n)}$ (see Notation 8.6); we claim:

(22) $\big{(}B_{{\rm\bf k};{\mathbf{a}}}/x_{n}B_{{\rm\bf
k};{\mathbf{a}}}\big{)}^{(d_{n})}\overset{\text{\tiny$*$}}{\cong}\big{(}B_{{\rm\bf
k};{\mathbf{a}}(n)}\big{)}^{(d_{n})}.$

This has to be carefully argued, because in general the condition
$A\overset{\text{\tiny$*$}}{\cong}B$ does not imply that
$A^{(d)}\overset{\text{\tiny$*$}}{\cong}B^{(d)}$ for all $d$. If we use the
notations $\bar{B}=B_{{\rm\bf k};{\mathbf{a}}}/x_{n}B_{{\rm\bf
k};{\mathbf{a}}}$, $C=B_{{\rm\bf k};{\mathbf{a}}(n)}$ and
$\varepsilon=\gcd(d_{1},\dots,d_{n-1})$, then $e(\bar{B})=\varepsilon$ and
there is an isomorphism of ${\rm\bf k}$-algebras $\varphi:\bar{B}\to C$
satisfying:

$\varphi(\bar{B}_{i})=\begin{cases}C_{i/\varepsilon}&\text{if
$i\in\varepsilon\mathbb{N}$,}\\\ 0&\text{if
$i\in\mathbb{N}\setminus\varepsilon\mathbb{N}$.}\end{cases}$

Let $\ell=\operatorname{\text{\rm lcm}}(\varepsilon,d_{n})$; then
$e(\bar{B}^{(d_{n})})=\ell$ by Lemma 7.19(a), so
$\bar{B}^{(d_{n})}=\bar{B}^{(\ell)}$ and hence

$\textstyle\varphi\big{(}\bar{B}^{(d_{n})}\big{)}=\varphi\big{(}\bar{B}^{(\ell)}\big{)}=\varphi\big{(}\bigoplus_{i\in\ell\mathbb{N}}\bar{B}_{i}\big{)}=\bigoplus_{i\in\ell\mathbb{N}}C_{i/\varepsilon}=\bigoplus_{j\in(\ell/\varepsilon)\mathbb{N}}C_{j}=C^{(\ell/\varepsilon)},$

showing that
$(\bar{B})^{(d_{n})}\overset{\text{\tiny$*$}}{\cong}C^{(\ell/\varepsilon)}$.
Since $\gcd(\varepsilon,d_{n})=\gcd(d_{1},\dots,d_{n})=1$, we have
$\ell/\varepsilon=d_{n}$, so
$(\bar{B})^{(d_{n})}\overset{\text{\tiny$*$}}{\cong}C^{(d_{n})}$. This proves
(22). Next, we prove:

(23) $\gcd(d_{n},\bar{e}(B_{{\rm\bf k};{\mathbf{a}}(n)}))=1.$

We have $\bar{e}(B_{{\rm\bf
k};{\mathbf{a}}(n)})=\prod_{j=1}^{n-1}\frac{\operatorname{\text{\rm
lcm}}(a_{1},\dots,a_{n-1})}{\operatorname{\text{\rm
lcm}}(a_{1},\dots,\widehat{a_{j}},\dots,a_{n-1})}$ by Lemma 8.7. Suppose that
$p$ is a prime number that divides both $d_{n}$ and $\bar{e}(B_{{\rm\bf
k};{\mathbf{a}}(n)})$. Since
$p\mid\prod_{j=1}^{n-1}\frac{\operatorname{\text{\rm
lcm}}(a_{1},\dots,a_{n-1})}{\operatorname{\text{\rm
lcm}}(a_{1},\dots,\widehat{a_{j}},\dots,a_{n-1})}$, there exists
$j\in\\{1,\dots,n-1\\}$ such that
$v_{p}(a_{j})>\max(v_{p}(a_{1}),\dots,\widehat{v_{p}(a_{j})},\dots,v_{p}(a_{n-1}))$.
Since $p$ divides $d_{n}=L/a_{n}$, we have
$v_{p}(a_{n})<\max(v_{p}(a_{1}),\dots,v_{p}(a_{n-1}))=v_{p}(a_{j})$, so
$v_{p}(a_{j})>\max(v_{p}(a_{1}),\dots,\widehat{v_{p}(a_{j})},\dots,v_{p}(a_{n}))=v_{p}(L_{j})$.
It follows that $v_{p}(L)\geq v_{p}(a_{j})>v_{p}(L_{j})$, so $j\in
S({\mathbf{a}})$, which contradicts $S({\mathbf{a}})\subseteq\\{n\\}$. So $p$
does not exist, and this proves (23).

Since $\big{(}B_{{\rm\bf k};{\mathbf{a}}}/x_{n}B_{{\rm\bf
k};{\mathbf{a}}}\big{)}^{(d_{n})}$ is non-rigid, (22) implies that
$(B_{{\rm\bf k};{\mathbf{a}}(n)})^{(d_{n})}$ is non-rigid. Thus, $d_{n}\in
X(B_{{\rm\bf k};{\mathbf{a}}(n)})=\bigcup_{d\in M(B_{{\rm\bf
k};{\mathbf{a}}(n)})}\mathscr{I}_{d}$, where the equality is Cor. 7.14(a). It
follows that there exists $d\in M(B_{{\rm\bf k};{\mathbf{a}}(n)})$ such that
$d\mid d_{n}$. Cor. 7.14(b) gives $d\mid\bar{e}(B_{{\rm\bf
k};{\mathbf{a}}(n)})$, so $d=1$ by (23). This means that $1\in M(B_{{\rm\bf
k};{\mathbf{a}}(n)})$, so $B_{{\rm\bf k};{\mathbf{a}}(n)}$ is non-rigid, as
desired. ∎

######  9.9 Example.

Let ${\rm\bf k}$ be a field of characteristic $0$, let
$a,b\in\mathbb{N}\setminus\\{0,1,2\\}$, and consider $B=B_{{\rm\bf
k};a,a,b,b}$. We claim that

$B_{(x_{i})}$ is rigid for every $i\in\\{1,2,3,4\\}$.

Indeed, for each $i\in\\{1,2,3,4\\}$ we have $B_{{\rm\bf
k};{\mathbf{a}}(i)}=B_{{\rm\bf k};a,b,b}$ or $B_{{\rm\bf k};a,a,b}$, so
$B_{{\rm\bf k};{\mathbf{a}}(i)}$ is rigid by Lemma 8.5; since
$S(a,a,b,b)=\varnothing\subseteq\\{i\\}$, Prop. 9.8 implies that $B_{(x_{i})}$
is rigid.

##  10\. Fibers of a homogeneous polynomial

The title of this section refers to Prop. 10.5, which is the main result of
the section.

######  10.1 Lemma ([Sch00], Chap. 3, § 3, Cor. 1).

Let ${\rm\bf k}$ be an algebraically closed field, $n\geq 1$ and $F\in
A={\rm\bf k}^{[{n}]}$. Then the set $\big{\\{}\,\lambda\in{\rm\bf
k}\,\mid\,\text{$F-\lambda$ is not irreducible in $A$}\,\big{\\}}$ is either
finite or equal to ${\rm\bf k}$, and it is equal to ${\rm\bf k}$ if and only
if $F=P(G)$ for some $G\in A$ and some univariate polynomial $P(T)\in{\rm\bf
k}[T]$ such that $\deg_{T}P(T)>1$.

######  10.2 Lemma (Lemma 6.1 of [Dai20]).

Let $A$ be an algebra over a field ${\rm\bf k}$, let $K/{\rm\bf k}$ be an
algebraic Galois extension and write $G=\operatorname{\text{\rm
Gal}}(K/{\rm\bf k})$ and $A_{K}=K\otimes_{\rm\bf k}A$. For each $\theta\in G$,
let $\tilde{\theta}:A_{K}\to A_{K}$ be the $A$-automorphism of $A_{K}$ given
by $\tilde{\theta}(\lambda\otimes a)=\theta(\lambda)\otimes a$ ($\lambda\in
K$, $a\in A$). If $b$ is an element of $A_{K}$ satisfying

(24) $\forall_{\theta\in G}\ \exists_{\lambda\in K^{*}}\
\tilde{\theta}(b)=\lambda b$

then there exists $\lambda\in K^{*}$ such that $\lambda b\in A$.

######  10.3 Lemma.

Let ${\rm\bf k}$ be a field of characteristic $0$, $\bar{{\rm\bf k}}$ an
algebraic closure of ${\rm\bf k}$, $n\geq 1$ and $R={\rm\bf
k}[X_{1},\dots,X_{n}]={\rm\bf k}^{[n]}$. Endow $R$ with an
$\mathbb{N}$-grading such that each $X_{i}$ is homogeneous and let $f$ be a
homogeneous prime element of $R$ of positive degree. Then $f-\lambda$ is
irreducible over $\bar{{\rm\bf k}}$ for every $\lambda\in\bar{{\rm\bf
k}}^{*}$.

###### Proof.

Let $\bar{R}=\bar{{\rm\bf k}}[X_{1},\dots,X_{n}]=\bar{{\rm\bf k}}^{[n]}$. Let
$m=\deg(f)>0$ and $d_{i}=\deg(X_{i})$ ($1\leq i\leq n$). For each
$\lambda\in\bar{{\rm\bf k}}^{*}$, consider the $\bar{{\rm\bf k}}$-automorphism
$\mu_{\lambda}$ of $\bar{R}$ defined by
$\mu_{\lambda}(X_{i})=\lambda^{d_{i}}X_{i}$ ($1\leq i\leq n$). If
$\lambda,\lambda_{0}\in\bar{{\rm\bf k}}^{*}$ then
$\mu_{\lambda}(f-\lambda^{m}\lambda_{0})=\lambda^{m}(f-\lambda_{0})$, so the
$\bar{{\rm\bf k}}$-algebras $\bar{R}/(f-\lambda_{0})$ and
$\bar{R}/(f-\lambda^{m}\lambda_{0})$ are isomorphic. Since $m>0$, it follows
that

(25) for all $\lambda_{1},\lambda_{2}\in\bar{{\rm\bf k}}^{*}$,
$\bar{R}/(f-\lambda_{1})$ is $\bar{{\rm\bf k}}$-isomorphic to
$\bar{R}/(f-\lambda_{2})$.

Arguing by contradiction, suppose that $f-\lambda$ is reducible in $\bar{R}$,
for some $\lambda\in\bar{{\rm\bf k}}^{*}$. Then, by (25), $f-\lambda$ is
reducible in $\bar{R}$ for all $\lambda\in\bar{{\rm\bf k}}^{*}$. So Lemma 10.1
implies that $f=P(g)$ for some $g\in\bar{R}$ and some univariate polynomial
$P(T)\in\bar{{\rm\bf k}}[T]$ such that $\deg_{T}P(T)>1$. Using that $f$ is
homogeneous of positive degree, we see that $g$ may be chosen so as to have
$P(T)=T^{r}$, where $r\geq 2$. Thus, $f=g^{r}$. For each
$\theta\in\operatorname{\text{\rm Gal}}(\bar{{\rm\bf k}}/{\rm\bf k})$, define
$\tilde{\theta}:\bar{R}\to\bar{R}$ as in Lemma 10.2. Then
$\tilde{\theta}(g)^{r}=f$, so $(\tilde{\theta}(g)/g)^{r}=1$. This shows that
for each $\theta\in\operatorname{\text{\rm Gal}}(\bar{{\rm\bf k}}/{\rm\bf k})$
there exists $\lambda_{\theta}\in\bar{{\rm\bf k}}^{*}$ such that
$\tilde{\theta}(g)=\lambda_{\theta}g$. By Lemma 10.2, it follows that there
exists $\lambda\in\bar{{\rm\bf k}}^{*}$ such that $\lambda g\in R$. So
$f=(\lambda^{-r})(\lambda g)^{r}$ is not irreducible in $R$, a contradiction.
∎

######  10.4 Lemma.

Let ${\rm\bf k}$ be a field of characteristic $0$, $\bar{{\rm\bf k}}$ an
algebraic closure of ${\rm\bf k}$, $n\geq 2$, and $f$ an irreducible element
of $R={\rm\bf k}[X_{1},\dots,X_{n}]={\rm\bf k}^{[{n}]}$. Then the zero-set of
$(f,\frac{\partial f}{\partial X_{1}},\dots,\frac{\partial f}{\partial
X_{n}})$ in $\mathbb{A}^{n}_{\bar{{\rm\bf k}}}$ has dimension at most $n-2$.

###### Proof.

If the conclusion is false then there exists a prime element $g$ of
$\bar{R}=\bar{{\rm\bf k}}[X_{1},\dots,X_{n}]$ which is a common divisor of
$f$, $\frac{\partial f}{\partial X_{1}}$, …, $\frac{\partial f}{\partial
X_{n}}$ in $\bar{R}$. Choose $i$ such that $\frac{\partial g}{\partial
X_{i}}\neq 0$ and note that $\frac{\partial f}{\partial X_{i}}\neq 0$. Let
$K={\rm\bf k}(X_{1},\dots,\widehat{X_{i}},\dots,X_{n})$ and
$\bar{K}=\bar{{\rm\bf k}}(X_{1},\dots,\widehat{X_{i}},\dots,X_{n})$. Note that
$f$ (resp. $g$) is an irreducible element of $K[X_{i}]=K^{[1]}$ (resp. of
$\bar{K}[X_{i}]=\bar{K}^{[1]}$). It follows that $\gcd(f,\frac{\partial
f}{\partial X_{i}})=1$ in $K[X_{i}]$, so $uf+v\frac{\partial f}{\partial
X_{i}}=1$ for some $u,v\in K[X_{i}]$, so $\gcd(f,\frac{\partial f}{\partial
X_{i}})=1$ in $\bar{K}[X_{i}]$, contradicting the fact that $g$ is an
irreducible element of $\bar{K}[X_{i}]$ that divides both $f$ and
$\frac{\partial f}{\partial X_{i}}$. ∎

######  10.5 Proposition.

Let ${\rm\bf k}$ be a field of characteristic $0$ and $R={\rm\bf
k}[X_{1},\dots,X_{n}]={\rm\bf k}^{[{n}]}$, where $n\geq 2$. Endow $R$ with an
$\mathbb{N}$-grading such that each $X_{i}$ is homogeneous and $X_{1}$ has
positive degree. Let $f$ be a homogeneous prime element of
$R=\bigoplus_{i\in\mathbb{N}}R_{i}$ such that $f\notin R_{0}[X_{1}]$.

  1. (a)

$R/(f-c)R$ is a domain, for every $c\in{\rm\bf k}^{*}$.

  2. (b)

If $R/fR$ is rigid then so is $R/(f-c)R$ for every $c\in{\rm\bf k}^{*}$.

###### Proof.

Part (a) follows from Lemma 10.3.

Let $c\in{\rm\bf k}^{*}$. Let $m=\deg(f)>0$ and
$d_{i}=\deg(X_{i})\in\mathbb{N}$ ($1\leq i\leq n$). We claim that
$f-cX_{n+1}^{m}$ is irreducible in $\bar{{\rm\bf
k}}[X_{1},\dots,X_{n+1}]=\bar{{\rm\bf k}}^{[n+1]}$, where $\bar{{\rm\bf k}}$
is an algebraic closure of ${\rm\bf k}$. Indeed, suppose that
$G,H\in\bar{{\rm\bf k}}[X_{1},\dots,X_{n+1}]$ are such that
$f-cX_{n+1}^{m}=GH$. For each $\lambda\in\bar{{\rm\bf k}}^{*}$, define
$G(\lambda)=G(X_{1},\dots,X_{n},\lambda)$ and
$H(\lambda)=H(X_{1},\dots,X_{n},\lambda)$. Then for each
$\lambda\in\bar{{\rm\bf k}}^{*}$ we have
$f-c\lambda^{m}=G(\lambda)H(\lambda)$. Since $f-c\lambda^{m}$ is irreducible
over $\bar{{\rm\bf k}}$ by Lemma 10.3, it follows that
$G(\lambda)\in\bar{{\rm\bf k}}^{*}$ or $H(\lambda)\in\bar{{\rm\bf k}}^{*}$. So
there exists $W\in\\{G,H\\}$ such that $W(\lambda)\in\bar{{\rm\bf k}}^{*}$ is
true for infinitely many $\lambda\in\bar{{\rm\bf k}}$. This implies that
$W\in\bar{{\rm\bf k}}[X_{n+1}]$. There exists $V\in\\{G,H\\}$ such that
$WV=f-cX_{n+1}^{m}$; pick $i\in\\{1,\dots,n\\}$ such that $\frac{\partial
f}{\partial X_{i}}\neq 0$, then $W\frac{\partial V}{\partial
X_{i}}=\frac{\partial f}{\partial X_{i}}\in{\rm\bf
k}[X_{1},\dots,X_{n}]\setminus\\{0\\}$ shows that $W\in\bar{{\rm\bf k}}^{*}$,
which, in turn, shows that $f-cX_{n+1}^{m}$ is irreducible in $\bar{{\rm\bf
k}}[X_{1},\dots,X_{n+1}]$. So both $B={\rm\bf
k}[X_{1},\dots,X_{n+1}]/(f-cX_{n+1}^{m})$ and $\bar{B}=\bar{{\rm\bf
k}}\otimes_{\rm\bf k}B$ are domains.

We claim that $B$ and $\bar{B}$ are normal. This is clear if $m=1$, so assume
that $m\geq 2$. Then the singular locus of the affine variety
$X=V(f-cX_{n+1}^{m})\subset\mathbb{A}_{\bar{{\rm\bf k}}}^{n+1}$ is
$V(X_{n+1},f,\frac{\partial f}{\partial X_{1}},\dots,\frac{\partial
f}{\partial X_{n}})\subseteq\mathbb{A}_{\bar{{\rm\bf k}}}^{n+1}$. By Lemma
10.4, we see that the codimension of $\operatorname{\text{\rm Sing}}(X)$ in
$X$ is greater than $1$; since $X$ is a hypersurface of
$\mathbb{A}^{n+1}_{\bar{{\rm\bf k}}}$, it follows that $X$ is normal. Since
$X\cong\operatorname{\text{\rm Spec}}\bar{B}$, $\bar{B}$ is normal. It is well
known that $\bar{B}\cap\operatorname{\text{\rm Frac}}(B)=B$, so $B$ is normal.

Define an $\mathbb{N}$-grading on ${\rm\bf k}[X_{1},\dots,X_{n+1}]$ by
declaring that $X_{n+1}$ is homogeneous of degree $1$ and (for $1\leq i\leq
n$) $X_{i}$ is homogeneous of degree $d_{i}$. Then $f-cX_{n+1}^{m}$ is
homogeneous, so $B$ is $\mathbb{N}$-graded. Write $B={\rm\bf
k}[x_{1},\dots,x_{n+1}]=\bigoplus_{i\in\mathbb{N}}B_{i}$ where $x_{i}$ is the
canonical image of $X_{i}$ in $B$. Since $x_{1}$ and $x_{n+1}$ have positive
degrees and are algebraically independent over $B_{0}$ (because
$f-cX_{n+1}^{m}\notin R_{0}[X_{1},X_{n+1}]$), $B$ has transcendence degree at
least $2$ over $B_{0}$; so the prime ideal $B_{+}=\bigoplus_{i>0}B_{i}$ has
height at least $2$. We showed:

$B$ is an $\mathbb{N}$-graded normal affine ${\rm\bf k}$-domain such that
$\operatorname{\text{\rm ht}}(B_{+})\geq 2$.

Since $B/x_{n+1}B\cong R/fR$ is a domain, $x_{n+1}$ is a homogeneous prime
element of $B$ of degree $1$. We claim that

(26) $B_{(x_{n+1})}\cong R/(f-c).$

To see this, let $\varphi:R\to B_{(x_{n+1})}$ be the surjective ${\rm\bf
k}$-homomorphism defined by $\varphi(X_{i})=x_{i}/x_{n+1}^{d_{i}}$ ($1\leq
i\leq n$). We have
$\varphi(f)=f\big{(}{x_{1}}/{x_{n+1}^{d_{1}}},\dots,{x_{n}}/{x_{n+1}^{d_{n}}}\big{)}=f/x_{n+1}^{m}=c$,
so $(f-c)\subseteq\ker\varphi$. Lemma 10.3 implies that $f-c$ is a prime
element of $R$, so $(f-c)$ is a height $1$ prime ideal of $R$. Since $\dim
R=n$ and $\dim B_{(x_{n+1})}=\dim\operatorname{\text{\rm Proj}}B=n-1$,
$\ker\varphi$ is a height $1$ prime ideal of $R$, so $\ker\varphi=(f-c)$,
which proves (26).

Let us apply Thm 9.6 to $B$ and $x=x_{n+1}$ (note that $d=\deg(x)=1$). By that
result and the fact that $(B/xB)^{(d)}=B/xB=B/x_{n+1}B\cong R/fR$ is rigid by
assumption, we obtain that $B_{(x_{n+1})}$ is rigid, so $R/(f-c)$ is rigid by
(26). ∎

######  10.6 Example.

Let $A=\mathbb{C}[X,Y,Z]/(X^{2}+Y^{3}+Z^{3}-1)$. One way to prove that $A$ is
rigid is to apply Prop. 10.5 to $f=X^{2}+Y^{3}+Z^{3}\in R=\mathbb{C}[X,Y,Z]$,
where $X,Y,Z$ are homogeneous of degrees $3,2,2$. Since $R/fR\cong
B_{\mathbb{C};2,3,3}$ is rigid by Lemma 8.5, the Proposition implies that $A$
is a rigid domain. (Remark: one can see that $\operatorname{\text{\rm Spec}}A$
is a smooth rational surface.)

##  11\. A result about $\operatorname{\bar{}\mathbb{G}}(B)$

The aim of this section is to prove Thm 11.4.

Given a subgroup $H$ of an abelian group $G$, we define the torsion-closure of
$H$ in $G$ to be $\operatorname{\text{\rm torc}}(H,G)=\big{\\{}\,x\in
G\,\mid\,\text{there exists $m\in\mathbb{N}\setminus\\{0\\}$ such that $mx\in
H$}\,\big{\\}}$.

######  11.1 Lemma.

Let $H$ be a subgroup of a finitely generated abelian group $G$ such that
$G/H$ is not torsion. Let $\mathcal{H}(H,G)$ be the set of all subgroups
$H^{\prime}$ of $G$ such that $H\subseteq H^{\prime}\subset G$ and
$G/H^{\prime}\cong\mathbb{Z}$. Then $\mathcal{H}(H,G)\neq\varnothing$ and the
intersection of all elements of $\mathcal{H}(H,G)$ is $\operatorname{\text{\rm
torc}}(H,G)$.

###### Proof.

Viewing $G$ as a $\mathbb{Z}$-module, let $S=\mathbb{Z}\setminus\\{0\\}$ and
consider the $\mathbb{Q}$-vector space $G_{\mathbb{Q}}=S^{-1}G$ and the
canonical homomorphism $\lambda:G\to G_{\mathbb{Q}}$, $x\mapsto x/1$. For each
subgroup $K$ of $G$, let $K_{\mathbb{Q}}$ be the subspace $S^{-1}K$ of
$G_{\mathbb{Q}}$ and note that
$\lambda^{-1}(K_{\mathbb{Q}})=\operatorname{\text{\rm torc}}(K,G)$.

Since $G/H$ is not torsion, there exists $y\in
G\setminus\operatorname{\text{\rm torc}}(H,G)$. Consider any such $y$.
Consider the vector $\lambda(y)\in G_{\mathbb{Q}}$ and note that
$\lambda(y)\notin H_{\mathbb{Q}}$. Choose a vector subspace $V_{y}$ of
$G_{\mathbb{Q}}$ such that $H_{\mathbb{Q}}\subseteq V_{y}\subset
G_{\mathbb{Q}}$, $\dim_{\mathbb{Q}}(G_{\mathbb{Q}}/V_{y})=1$, and
$\lambda(y)\notin V_{y}$. Let $H_{y}=\lambda^{-1}(V_{y})$. Then $y\notin
H_{y}$ and $H\subseteq H_{y}\subset G$; since $G/H_{y}$ is torsion-free (and
hence free, since $G$ is finitely generated), and since $S^{-1}(G/H_{y})\cong
G_{\mathbb{Q}}/V_{y}\cong\mathbb{Q}$, we have $G/H_{y}\cong\mathbb{Z}$; so
$H_{y}\in\mathcal{H}(H,G)$. This shows that $\mathcal{H}(H,G)\neq\varnothing$
and $y\notin I$, where $I$ denotes the intersection of all elements of
$\mathcal{H}(H,G)$. This being true for each $y\in
G\setminus\operatorname{\text{\rm torc}}(H,G)$, we obtain
$I\subseteq\operatorname{\text{\rm torc}}(H,G)$, and the reverse inclusion is
clear. ∎

######  11.2 Lemma.

Let $H_{1},\dots,H_{n}$ be subgroups of an abelian group $G$. Then

$\operatorname{\text{\rm torc}}(H_{1},G)\cap\dots\cap\operatorname{\text{\rm
torc}}(H_{n},G)=\operatorname{\text{\rm torc}}(H_{1}\cap\dots\cap H_{n},G).$

We leave it to the reader to verify Lemma 11.2.

######  11.3.

Let $B=\bigoplus_{i\in G}B_{i}$ be a domain graded by an abelian group $G$ and
let $H$ be a subgroup of $\operatorname{\mathbb{G}}(B)$. Let
$\Omega=\operatorname{\mathbb{G}}(B)/H$ and let
$\pi:\operatorname{\mathbb{G}}(B)\to\Omega$ be the canonical homomorphism of
the quotient group. Let $\mathcal{B}$ denote the ring $B$ endowed with the
$\Omega$-grading defined by
$\mathcal{B}=\bigoplus_{\omega\in\Omega}\mathcal{B}_{\omega}$, where
$\mathcal{B}_{\omega}=\bigoplus_{\pi(i)=\omega}B_{i}$ for each
$\omega\in\Omega$. We refer to this as “the natural
$\operatorname{\mathbb{G}}(B)/H$-grading”. It satisfies
$\operatorname{\mathbb{G}}(\mathcal{B})=\Omega$. Observe that
$\operatorname{\text{\rm Spec}}^{1}(\mathcal{B})=\operatorname{\text{\rm
Spec}}^{1}(B)$ and let us show that

(27)
$\pi\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}})\big{)}=\overline{\mathbb{M}}(\mathcal{B},{\mathfrak{p}})\quad\text{for
every ${\mathfrak{p}}\in\operatorname{\text{\rm Spec}}^{1}(B)$.}$

Indeed, let ${\mathfrak{p}}\in\operatorname{\text{\rm Spec}}^{1}(B)$. If
$i\in{\mathbb{M}}(B,{\mathfrak{p}})$ then $B_{i}\nsubseteq{\mathfrak{p}}$, so
$\mathcal{B}_{\pi(i)}\nsubseteq{\mathfrak{p}}$ (because $B_{i}\subseteq
B_{\pi(i)}$) and hence $\pi(i)\in\mathbb{M}(\mathcal{B},{\mathfrak{p}})$.
Thus,
$\pi\big{(}{\mathbb{M}}(B,{\mathfrak{p}})\big{)}\subseteq\mathbb{M}(\mathcal{B},{\mathfrak{p}})$.
Conversely, consider $\omega\in\mathbb{M}(\mathcal{B},{\mathfrak{p}})$. Then
$\mathcal{B}_{\omega}=\bigoplus_{\pi(i)=\omega}B_{i}$ is not included in
${\mathfrak{p}}$, so there exists $i\in\operatorname{\mathbb{G}}(B)$ such that
$\pi(i)=\omega$ and $B_{i}\nsubseteq{\mathfrak{p}}$; then
$i\in{\mathbb{M}}(B,{\mathfrak{p}})$ and hence
$\omega\in\pi\big{(}{\mathbb{M}}(B,{\mathfrak{p}})\big{)}$. This shows that
$\pi\big{(}{\mathbb{M}}(B,{\mathfrak{p}})\big{)}=\mathbb{M}(\mathcal{B},{\mathfrak{p}})$,
and it follows that
$\pi\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}})\big{)}=\overline{}\mathbb{M}(\mathcal{B},{\mathfrak{p}})$.
So (27) is proved. Caution: Assertion (27) implies that
$\pi\big{(}\operatorname{\bar{}\mathbb{G}}(B)\big{)}\subseteq\operatorname{\bar{}\mathbb{G}}(\mathcal{B})$,
but equality does not necessarily hold.

The rank of an abelian group $G$ is the dimension of the $\mathbb{Q}$-vector
space $\mathbb{Q}\otimes_{\mathbb{Z}}G$. See Rem. 3.5 for the definition of
$\operatorname{\text{\rm ML}}(B)$.

######  11.4 Theorem.

Let ${\rm\bf k}$ be a field of characteristic $0$ and $B$ a normal affine
${\rm\bf k}$-domain graded by an abelian group $G$. Let
$r=\operatorname{\text{\rm
rank}}\big{(}\operatorname{\mathbb{G}}(B)/\operatorname{\bar{}\mathbb{G}}(B)\big{)}$.

  1. (a)

$\operatorname{\text{\rm ML}}(B)\subseteq B^{(W)}$, where
$W=\operatorname{\text{\rm
torc}}(\operatorname{\bar{}\mathbb{G}}(B),\operatorname{\mathbb{G}}(B))$.

  2. (b)

There exists a field $K$ such that $\operatorname{\text{\rm ML}}(B)\subseteq
K\subseteq\operatorname{\text{\rm Frac}}(B)$ and $\operatorname{\text{\rm
Frac}}(B)=K^{(r)}$. In particular, $\operatorname{\text{\rm
trdeg}}(B:\operatorname{\text{\rm ML}}(B))\geq r$.

  3. (c)

If $B$ is rigid then
$\operatorname{\mathbb{G}}(B)/\operatorname{\bar{}\mathbb{G}}(B)$ is a finite
group.

###### Proof.

Since $B$ is ${\rm\bf k}$-affine, $\operatorname{\mathbb{G}}(B)$ is finitely
generated and consequently $r\in\mathbb{N}$.

(a) If $\operatorname{\text{\rm Spec}}^{1}(B)=\varnothing$ then
$\operatorname{\bar{}\mathbb{G}}(B)=\operatorname{\mathbb{G}}(B)$ by
definition, so (a) is claiming that $\operatorname{\text{\rm ML}}(B)\subseteq
B$, which is true. From now-on, assume that $\operatorname{\text{\rm
Spec}}^{1}(B)\neq\varnothing$.

By Prop. 4.4, the nonempty set
$\big{\\{}\,\overline{\mathbb{M}}(B,{\mathfrak{p}})\,\mid\,{\mathfrak{p}}\in\operatorname{\text{\rm
Spec}}^{1}(B)\,\big{\\}}$ is finite. So we can choose
${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}\in\operatorname{\text{\rm
Spec}}^{1}(B)$ such that
$\operatorname{\bar{}\mathbb{G}}(B)=\bigcap_{i=1}^{n}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i})$.
By Lemma 11.2 we have $\bigcap_{i=1}^{n}\operatorname{\text{\rm
torc}}\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i}),\operatorname{\mathbb{G}}(B)\big{)}=\operatorname{\text{\rm
torc}}(\operatorname{\bar{}\mathbb{G}}(B),\operatorname{\mathbb{G}}(B))$,
i.e.,

(28) $\textstyle\bigcap_{i=1}^{n}\operatorname{\text{\rm
torc}}\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i}),\operatorname{\mathbb{G}}(B)\big{)}=W.$

Let $I=\big{\\{}\,i\,\mid\,\text{$1\leq i\leq n$ and
$\operatorname{\mathbb{G}}(B)/\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i})$ is
not torsion}\,\big{\\}}$. If $I=\varnothing$ then (28) implies that
$W=\operatorname{\mathbb{G}}(B)$, in which case the assertion of (a) is
“$\operatorname{\text{\rm ML}}(B)\subseteq B$”, which is true. So we may
assume that $I\neq\varnothing$. We have $\operatorname{\text{\rm
torc}}\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i}),\operatorname{\mathbb{G}}(B)\big{)}=\operatorname{\mathbb{G}}(B)$
for each $i\in\\{1,\dots,n\\}\setminus I$, so (28) implies that

(29) $\textstyle\bigcap_{i\in I}\operatorname{\text{\rm
torc}}\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i}),\operatorname{\mathbb{G}}(B)\big{)}=W.$

Define
$\mathcal{H}_{i}=\mathcal{H}\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i}),\operatorname{\mathbb{G}}(B)\big{)}$
for each $i\in I$. Lemma 11.1 implies that

for each $i\in I$, $\mathcal{H}_{i}\neq\varnothing$ and
$\textstyle\bigcap_{H\in\mathcal{H}_{i}}H=\operatorname{\text{\rm
torc}}\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i}),\operatorname{\mathbb{G}}(B)\big{)}$.

Consequently, $W=\bigcap_{i\in I}\bigcap_{H\in\mathcal{H}_{i}}H$. In other
words, if we define $\mathcal{H}=\bigcup_{i\in I}\mathcal{H}_{i}$, then

(30) $\textstyle W=\bigcap_{H\in\mathcal{H}}H.$

Let $H\in\mathcal{H}$. Let
$\pi:\operatorname{\mathbb{G}}(B)\to\operatorname{\mathbb{G}}(B)/H$ be the
canonical homomorphism and choose $i\in I$ such that $H\in\mathcal{H}_{i}$.
The definition of $\mathcal{H}_{i}$ implies that
$\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i})\subseteq H=\ker(\pi)$ and
$\operatorname{\mathbb{G}}(B)/H\cong\mathbb{Z}$. Let $\mathcal{B}(H)$ denote
the ring $B$ endowed with the natural $\operatorname{\mathbb{G}}(B)/H$-grading
(see paragraph 11.3) and note that
$\operatorname{\mathbb{G}}(\mathcal{B}(H))=\operatorname{\mathbb{G}}(B)/H\cong\mathbb{Z}$.

Since $\operatorname{\text{\rm Spec}}^{1}(B)=\operatorname{\text{\rm
Spec}}^{1}(\mathcal{B}(H))$, we have
${\mathfrak{p}}_{i}\in\operatorname{\text{\rm Spec}}^{1}(\mathcal{B}(H))$;
thus,

$\textstyle\operatorname{\bar{}\mathbb{G}}(\mathcal{B}(H))\ =\
\bigcap_{{\mathfrak{p}}\in\operatorname{\text{\rm
Spec}}^{1}(\mathcal{B}(H))}\overline{\mathbb{M}}(\mathcal{B}(H),{\mathfrak{p}})\
\subseteq\
\overline{\mathbb{M}}(\mathcal{B}(H),{\mathfrak{p}}_{i})=\pi\big{(}\overline{\mathbb{M}}(B,{\mathfrak{p}}_{i})\big{)}=0$

where the penultimate equality follows from (27). Since
$\operatorname{\mathbb{G}}(\mathcal{B}(H))\cong\mathbb{Z}$ and
$\operatorname{\bar{}\mathbb{G}}(\mathcal{B}(H))=0$, Cor. 7.18 implies that
there exists $D\in\operatorname{\text{\rm HLND}}(\mathcal{B}(H))$ such that
$\ker(D)=\mathcal{B}(H)_{0}=B^{(H)}$. Since $D\in\operatorname{\text{\rm
LND}}(B)$, we have $\operatorname{\text{\rm ML}}(B)\subseteq B^{(H)}$. This
being true for each $H\in\mathcal{H}$, we get $\operatorname{\text{\rm
ML}}(B)\subseteq\bigcap_{H\in\mathcal{H}}B^{(H)}=B^{(W)}$, the last equality
by (30). This proves (a).

(b) Let $R$ denote $B$ endowed with the natural
$\operatorname{\mathbb{G}}(B)/W$-grading (see paragraph 11.3). Since
$\operatorname{\mathbb{G}}(R)=\operatorname{\mathbb{G}}(B)/W$ is finitely
generated and torsion-free, and $\operatorname{\text{\rm
rank}}(\operatorname{\mathbb{G}}(B)/W)=\operatorname{\text{\rm
rank}}(\operatorname{\mathbb{G}}(B)/\operatorname{\bar{}\mathbb{G}}(B))=r\in\mathbb{N}$,
we have $\operatorname{\mathbb{G}}(R)\cong\mathbb{Z}^{r}$. Lemma 2.11 implies
that there exists a field $K$ such that $R_{0}\subseteq
K\subseteq\operatorname{\text{\rm Frac}}(R)$ and $\operatorname{\text{\rm
Frac}}(R)=K^{(r)}$. Since $R_{0}=B^{(W)}\supseteq\operatorname{\text{\rm
ML}}(B)$ by (a), we have $\operatorname{\text{\rm ML}}(B)\subseteq K$. Since
$\operatorname{\text{\rm Frac}}(B)=\operatorname{\text{\rm Frac}}(R)$, (b)
follows.

(c) Suppose that $B$ is rigid. Then (b) gives $\operatorname{\text{\rm
rank}}\big{(}\operatorname{\mathbb{G}}(B)/\operatorname{\bar{}\mathbb{G}}(B)\big{)}=0$,
so $\operatorname{\mathbb{G}}(B)/\operatorname{\bar{}\mathbb{G}}(B)$ is a
torsion group. Since $\operatorname{\mathbb{G}}(B)$ is finitely generated,
$\operatorname{\mathbb{G}}(B)/\operatorname{\bar{}\mathbb{G}}(B)$ is a finite
group. ∎

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# Bispectrum Analysis of Noninvasive EEG Signals Discriminates Complex and
Natural Grasp Types
††thanks: This research was partially supported by the RI-INBRE program (
through NIH P20GM103430) and NSF grant award ID 2245558.

Sima Ghafoori, Ali Rabiee, Anna Cetera, Reza Abiri  Department of Electrical,
Computer and Biomedical Engineering, University of Rhode Island, Kingston, RI,
USA
Emails: {sima.ghafoori, ali.rabiee, annacetera<EMAIL_ADDRESS>
###### Abstract

The bispectrum stands out as a revolutionary tool in frequency domain
analysis, leaping the usual power spectrum by capturing crucial phase
information between frequency components. In our innovative study, we have
utilized the bispectrum to analyze and decode complex grasping movements,
gathering EEG data from five human subjects. We put this data through its
paces with three classifiers, focusing on both magnitude and phase-related
features. The results highlight the bispectrum’s incredible ability to delve
into neural activity and differentiate between various grasping motions with
the Support Vector Machine (SVM) classifier emerging as a standout performer.
In binary classification, it achieved a remarkable 97% accuracy in identifying
power grasp, and in the more complex multiclass tasks, it maintained an
impressive 94.93% accuracy. This finding not only underscores the bispectrum’s
analytical strength but also showcases the SVM’s exceptional capability in
classification, opening new doors in our understanding of movement and neural
dynamics.

###### Index Terms:

EEG (Electroencephalograms), Bispectrum, Cross-bispectrum, Hand grip decoding,
and Machine Learning.

##  I Introduction

Reaching and grasping movements are imperative yet seemingly straightforward
daily activities to which we do not give much thought while performing them
[1, 2]. Even though prehension is done with ease, its underlying neural
activities have been proven to be complex and have been investigated using
various research techniques and methods[3, 4]. Moreover, studies have taken a
path to delve into the concept of grasping activities by deploying more
sophisticated analysis methodologies to detect and discriminate different
types of movements based on various modalities [5]. Their aim is not only to
study the background neuroscience but also to employ their findings for
controlling neuroprosthetic devices using human intentions [5]. Given that the
ultimate agenda is presenting a safe, portable, and affordable setup, there
has been rising interest in using Electroencephalography (EEG) as a prominent
biomarker for such research [6]. There has been an eclectic mix of techniques,
procedures, and platforms offered by articles for acquiring and studying
information about different reach-to-grasp movements [7, 8, 9, 10].
Furthermore, how they proceed with outlining their experimental setup is
diverse. Moreover, the studies have inspected the EEG signals using different
temporal and spectral analysis methods. The prominent method for analyzing
signals temporally is using Movement-Related Cortical Potential (MRCP)
components. In terms of spectral analysis, Power Density functions (PSD) have
a significant play alongside the less-used method of wavelet transform.
Moreover, few articles employed higher-order analytic methods such as
bispectrum to study imagined movements [11, 12]. To the best of our knowledge,
no articles looked into bispectrum for discrimination of motor executed reach-
to-grasp movements. Further, for distinguishing the defined reach-to-grasp
movements in articles, Machine Learning (ML) techniques such as Support Vector
Machines, Random Forest (RF), Linear Regression, and the most frequent one,
Linear Discriminant Analysis (LDA) have been employed [7, 8, 9, 10]. Hence, in
the current study, we opted for exploiting a state-of-the-art approach using
bispectrum to study the underlying neural activities behind two types of the
most common daily graspings, precision grip and power grip. Further, we
applied ML to distinguish the two movements. Hence, to acquire the informative
data, we designed a new platform and used pen and bottle for power and
precision grasps respectively. We also incorporated a space for no movement.

Figure 1: Representation of (a) the designed platform, (b) locations of EEG
electrodes, and (c) experimental setup

##  II Methods

###  II-A Platform Design and Experimental Paradigm

In the experimental setup, a 3D-designed, motorized turntable divided into
three sections was used to present two objects (a bottle and a pen) and a no-
object scenario. The turntable was operated by a computer, utilizing a TB6600
4A 9-42V Stepper Motor Driver and a Nema 17 Stepper Motor with 1.7A Bipolar.
Custom Python software was employed to ensure the synchronized functioning of
all hardware and software elements, as well as for logging events, which was
critical for processing EEG signals. Participants wore ”smart eyeglasses” that
were PC-controlled to switch between transparent and opaque states using a
smart film. This mechanism was key in controlling the visibility of objects to
the participants, ensuring their reactions were unbiased. During the
experiment, participants sat in a comfortable chair, maintaining a neutral
posture with their palms down and positioned 30 cm from the object’s center.
As shown in Figure 1c, after observing for 3 seconds, a buzzing sound was
emitted as an auditory cue, signaling the start of the grasping phase with
their dominant hand, constrained to 3 seconds. After each grasping phase, the
turntable moved on to the next object or the no-object condition. During this
transition, the smart eyeglasses were made opaque to avoid any anticipatory
bias in the participants. Figure 1 shows the designed platform and
experimental paradigm.

###  II-B Data Recording and Preprocessing

EEG measurements were obtained using the Unicorn Hybrid Black headset, a
mobile and wireless EEG tool with dry electrodes. It features eight electrodes
and records at a frequency of 250 Hz (more information can be found at
[Unicorn Hybrid Black headset](https://www.unicorn-bi.com). The electrode
arrangement is shown in Figure 1b. All electrodes used the mastoids as a
reference point, employing a linked mastoid reference system. For each
experimental condition, we carried out 50 trials per participant for each
condition. Our preprocessing steps included detrending, band-pass filtering
(1-40 Hz, order of 5), baseline correction using the signals collected during
the rotation of the turntable, Z scoring for normalizing the data, and
independent component Analysis (ICA) for artifact removal.

###  II-C Participants

In our study, we enlisted five healthy volunteers from the University of Rhode
Island, including two males and three females aged 22 to 36 years, all right-
handed. None had a history of neurological disorders. Before starting, each
participant received a detailed explanation of the experiment and gave written
consent. The research methods were approved by the Institutional Review Board
(IRB) to ensure they met ethical research standards.

###  II-D Bispectrum Analysis

The bispectrum is a higher-order frequency domain analysis tool that is
particularly useful in signal processing for identifying phase coupling and
non-linear interactions in signals. It is an extension of the power spectrum,
a second-order statistic, to the third-order. For a continuous signal $x(t)$,
the bispectrum $B(f_{1},f_{2})$ is defined as the triple product of the
Fourier transform of the signal at different frequencies. Mathematically, it
can be expressed as [11]:

$B(f_{1},f_{2})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}X(f_{1})X(f_{2})X^{*}(f_{1}+f_{2})\,df_{1}\,df_{2}$

where $X(f)$ is the Fourier transform of $x(t)$, and $X^{*}(f)$ is the complex
conjugate of $X(f)$, and $f_{1}$ and $f_{2}$ are frequency components. In this
study, Bispectrum was calculated using a 256-point Fast Fourier Transform
(FFT) with Hanning windows and 50 percent overlap for time segments.

###  II-E Feature Extraction and classification

We extracted a comprehensive set of features from the bispectrum of the whole
period of observation and execuation of each trial. It included both
amplitude-based and phase-related features. For the amplitude-based features,
the average bispectral power, the maximum bispectral power, the sum of
absolute bispectral power, the bispectral entropy, and the maximum bicoherence
we derived. These metrics collectively provided a detailed quantitative
assessment of the bispectrum’s power distribution and complexity.

In addition to these, we also delved into phase-related features, which were
crucial for understanding the phase relationships in the bispectrum.
Therefore, we computed the bispectral phase coherence in its real form, the
mean phase angle, and the second moment of the phase, offering insights into
the uniformity and variability of the phase data. Furthermore, phase-related
entropy and the real part of phase bicoherence were also added, adjusting the
calculation to account for normalization and providing a more nuanced
understanding of phase information. That left us with 80 features for each
trial (10 features from each channel). Using a 5-fold stratified cross-
validation approach, three classifiers (Random Forst: RF, Support Vector
Machin: SVM, Linear Discriminant Analysis: LDA) were trained and tested on our
dataset. It is worth noting that the training and testing sets were
standardized separately before the development of the models.

##  III Result

Figure 2: Average of bispectrum (top row) contour and 3D plots for C3 channel and cross-bispectrum (bottom row) magnitude plots for channels C3 and C4 TABLE I: 5-fold cross-validated accuracy scores for multiclass and binary classifications, reported for the 5 subjects (%). | Multiclass | Power Grasp vs No movement | Precision Grasp vs No movement | Power Grasp vs Precision Grasp
---|---|---|---|---
|  RF |  SVM |  LDA |  RF |  SVM |  LDA |  RF |  SVM |  LDA |  RF |  SVM |  LDA
S1 |  93.78 |  95.56 |  79.56 |  96.67 |  96.33 |  88.67 |  92.67 |  95.67 |  86.67 |  68.4 |  73 |  65.2
S2 |  93.55 |  93.12 |  81.11 |  96.33 |  96.67 |  94 |  93.33 |  94 |  86.33 |  72.33 |  69 |  67.08
S3 |  92.22 |  95.33 |  71.78 |  94.67 |  96.67 |  85 |  94 |  95.67 |  84.33 |  76 |  75 |  74
S4 |  94.67 |  94.89 |  81.78 |  95.67 |  98.67 |  92 |  95 |  96.11 |  85 |  71 |  68.86 |  69
S5 |  97.33 |  95.78 |  82.44 |  99 |  96.67 |  96.33 |  98.67 |  96.67 |  98 |  76.03 |  73.99 |  76
Mean |  94.31 |  94.93 |  79.33 |  96.47 |  97 |  91.2 |  94.73 |  95.62 |  88.06 |  72.75 |  71.97 |  70.25

Channels C3 and C4 are pivotal in capturing important information in motor
tasks [11, 9]. Therefore, the bispectrum of all trials was pooled together and
averaged for Channel C3 in three conditions. Figure 2 (top row) depicts the
contour and 3D plots of averaged bispectrum magnitudes for power, precision,
and no movements for frequencies less than 30 Hz for channel C3. The
associated images show relationships for both positive and negative
frequencies. The bottom row of images also represents the same concept but
between two channels of C3 and C4 called cross-bispectrum, which demonstrates
the frequency interaction between the two channels over the motor cortex. The
bispectrum magnitudes of the two types of movements are about 10 times more
than those related to the no-movement tasks. Also, the cross-bispectrum
magnitude plots seem to be more focused around certain frequencies less than
20 Hz.

Figure 3: Topography plots of Bispectrum Magnitude (top row) and phase (bottom
row) at 14 Hz and two-time points during Observation (2nd sec) and grasping
(4th sec) in three conditions

Table I shows the average accuracy scores of 5-fold cross-validation of the
three classifiers for the five subjects using all channels. The accuracies
were calculated in 4 classification categories from multiclass to three
conditions of binary classification. The power grasp seems to be
distinguishable more easily than the precision grasp since the scores obtained
from classifiers are notably more in classifying power grip compared to the
precision grip in binary classifications. Also, the data from subject 5
resulted in better results. Overall, SVM outperformed the other two
classifiers in most cases with an accuracy of 97% in power grip versus no-
movement classification and 94.93% in multiclass classification. Notably, the
models underperformed in power versus precision classification compared to
multiclass classification, with scores around 70%. Figure 3 shows topography
plots of Bispectrum Magnitude and phase at 14 Hz and two-time points, 2nd and
4th seconds, before and after auditory cue in three conditions. During
observation, we see higher values for bispectrum magnitude in premotor areas,
and during grasping we mostly see heightened activities in motor and occipital
areas. The same thing goes for phase interactions. The figure indicates how
neural interactions evolve from observation to execution at the frequency of
14 Hz which is within the beta band.

##  IV Discussion

In this study, we investigated the use of the bispectrum to distinguish
between two types of hand movements: power grasping and precision grasping. By
analyzing the EEG signals from the C3 channel and between the C3 and C4
channels, we observed distinct patterns in the frequency components of these
movements. During power and precision grasping, the bispectrum plots revealed
high-magnitude peaks, indicating strong non-linear interactions in the EEG
signals. These peaks were more spread out in power grasping, showing a range
of frequency interactions, while they were narrower and more defined in
precision grasping, suggesting specific frequency interactions during
controlled movements. In contrast, the no movement condition showed much lower
peaks, indicating less non-linear activity when the hand is at rest. These
findings highlight the brain’s adaptability and the varying complexities of
its electrical activity depending on the motor task.

Additionally, cross-bispectral analysis revealed phase coupling between
signals at different electrodes, particularly between C3 and C4 channels. This
analysis showed that power grasp involves a wider neural network, as seen in
the high-magnitude peaks, whereas precision grasp involves fewer but more
synchronized neural pathways. Furthermore, in both the bispectrum and cross-
bispectrum plots, power and precision grasps showed concentrated peaks, with
precision grasp displaying narrower peaks. The no-movement condition had
significantly reduced magnitudes, especially in the cross-bispectrum plots,
indicating minimal phase coupling due to lack of motor activity. Moreover, in
terms of classification, generally, RF and SVM performed better than LDA
across tasks, with SVM showing the highest mean accuracy in both multiclass
and binary classifications. The binary classification between power grasp and
no movement achieved the highest mean accuracy, suggesting that this
distinction is clearer in the EEG data. Overall, SVM results surpassed the
rest in most cases with a mean accuracy of 94.93% in multiclass, 97% in power
grip vs no-movement, 95.62% in precision grip vs no-movement, and 71.97% in
power grip vs precision grip classification. RF results were close to SVM and
LDA underperformed noticeably compared to the other two classifiers.

The machine learning methods applied in this study are widely recognized and
validated in numerous other research projects, and our results significantly
surpass the chance level of 33.33%, demonstrating robust and promising
outcomes. The power vs precision task has the lowest mean accuracy, indicating
that these two conditions are the hardest to differentiate for the
classifiers. These results can inform the development of more accurate models
for interpreting EEG data in applications such as brain-computer interfaces.
However, when comparing to the related work, the studies are different not
only in their analytic method of choice but also in the number of channels
(more than 32 electrodes), grip types, trials, and platform design [7, 8, 9,
10]. To the best of our knowledge, no study has ever investigated motor-
executed tasks with bispectrum. Studies mostly aim for imaginary tasks [11,
12] which require less sophisticated setups and platforms. Designing a
platform like this, which allows for ergonomic movements, presents a
significant challenge.

Lastly, the bispectral maps showed that observing an object and grasping it
activates different neural networks. Observation mainly activates visual and
attentional networks, while grasping also engages motor and sensorimotor
networks [1]. This is evident in the stronger magnitude values and specific
phase relationships in frequencies related to motor control during grasping,
compared to observation. This suggests different neural mechanisms are at play
during these tasks, although further analysis is needed to fully understand
these activities and interactions.

###  IV-A Limitations and Future Work

Bispectrum is a higher-order analysis method with so much potential for
exploration. For further research, it is recommended to delve into different
features and their contribution to classification tasks, to probe phase
information of the bispectrum more deeply. We just plot the cross-bispectrum
for two specific channels, there is definitely more room to look into that.
Also, the phase interactions between frequency bands can be studied in such
areas using bispectrum. It is worth noting that the number of channels and
trials was limited here. Also, there are other ML models worth evaluating
specifically in order to reach better results in discrimination between power
and precision movements.

##  V Conclusion

In our study, we focused on using the bispectrum method to understand brain
activity during reach-to-grasp movements. We found that the bispectrum gave us
important new insights into how the brain works during these tasks. The SVM
model, which used data from the bispectrum, was particularly good at
discriminating different kinds of movements.

## References

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