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Let $C$ be the following curve in $\mathbb{C}^2$. \begin{align} & 11664\, {c_1}^3\, {c_2}^2 + 536544\, {c_1}^3\, c_2 + 6170256\, {c_1}^3 + 67068\, {c_1}^2\, {c_2}^2 + 1542564\, {c_1}^2\, c_2 \\ & + 30 …

How to check that whether or not a surface in $\mathbb{C}^3$ is a K3 surface? Are there some method or math software to transform the equation of a surface to a normal form? For example, the following …

Let $\tilde{Gr}(k,n)$ be the affine cone of the Grassmannian $Gr(k,n)$. I think that the following set $S$ is an additive basis of $\mathbb{C}[\tilde{Gr}(k,n)]$: \begin{align} S = \{e_T: T \text{ is …

Let $R$ be a ring and $R((x))$ the ring of formal Laurent series. The elements in the ring $R((x))$ are series of the form $$ f = \sum_{n\in\mathbb{Z}} a_n x^n, $$ where ${\displaystyle a_{n}=0}$ for …

Let $f= xy^3+y^4-x^2+xy$. Using the following codes in Maple, f := xy^3+y^4-x^2+xy; v := Weierstrassform(f, x, y, x0, y0); I obtain the following result: \begin{align} & f_0 = {{ x_0}}^{3}+{{ y_ …

Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel …

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} P_{i …

Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. Then $Gr(k,n)$ is a projective variety and it has Plucker coordinates $P_{i_1, \ldots, i_k}$ ($i_1<\ldots< …

F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to clust …

I need to solve the following system of equations. solve([ a*c*e=a2*c2*e2, b*d*f=t*b2*d2*f2, t*a1*c1*e1=a3*c3*e3, b1*d1*f1=b3*d3*f3, a*b*a1*b1=a2*b2*a3*b3, c*d*c1*d1=c2*d2*c3*d3, e*f*e1*f1=e2*f2*e3* …

Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ and $I$ a non-homogeneous ideal of $R$. Then the algebra R/I is filtered. It has an associated graded algebra gr(R/I). Let $I_1$ be the initial ideal of $I$, …

In THE SLOPES DETERMINED BY n POINTS IN THE PLANE by JEREMY L. MARTIN, Page 2, Theorem 1.1, a Hilbert series is used to compute some combinatorial objects: Let $R_n=k[m_{1,2}, \ldots, m_{n-1,n}]$, a …

Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates) \begin{align} & {p_1}^2 {p_2}^2 c_1 {t_1}^2 t_2 + …

Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$: $$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and …

Let $\mathbb{G}_m$ be the multiplicative group and $T$ a maximal torus of a semisimple group. Let $X^*(T)=\{ \phi: T \to \mathbb{G}_m \}$ be the set of characters and $X_*(T)=\{ \phi^{\vee}: \mathbb{G …