Podcast
Questions and Answers
Consider a rational parametrization where $x = \frac{f_1(t_1, ..., t_m)}{g_1(t_1, ..., t_m)}, ..., x_n = \frac{f_n(t_1, ..., t_m)}{g_n(t_1, ..., t_m)}$. The map $\varphi : \mathbb{K}^m \rightarrow \mathbb{K}^n$ is not defined on all of $\mathbb{K}^m$. What condition defines the subset $W \subset \mathbb{K}^m$ where the map is undefined?
Consider a rational parametrization where $x = \frac{f_1(t_1, ..., t_m)}{g_1(t_1, ..., t_m)}, ..., x_n = \frac{f_n(t_1, ..., t_m)}{g_n(t_1, ..., t_m)}$. The map $\varphi : \mathbb{K}^m \rightarrow \mathbb{K}^n$ is not defined on all of $\mathbb{K}^m$. What condition defines the subset $W \subset \mathbb{K}^m$ where the map is undefined?
- $W = V(f_1 + f_2 + ... + f_n)$
- $W = V(g_1 + g_2 + ... + g_n)$
- $W = V(f_1 f_2 ... f_n)$
- $W = V(g_1 g_2 ... g_n)$ (correct)
In the context of rational implicitization, why is it necessary to add an extra dimension when dealing with rational parametrizations?
In the context of rational implicitization, why is it necessary to add an extra dimension when dealing with rational parametrizations?
- To ensure that the resulting variety is irreducible.
- To avoid having denominators vanish, effectively extending the domain of the parametrization. (correct)
- To reduce the degree of the polynomials involved.
- To simplify the polynomial equations involved in the implicitization process directly.
Given the ideal $J = \langle g_1x_1 - f_1, ..., g_nx_n - f_n, 1 - gy \rangle \subset \mathbb{K}[y, t_1, ..., t_m, x_1, ..., x_n]$, what is the significance of the term 1 - gy in the construction of $J$?
Given the ideal $J = \langle g_1x_1 - f_1, ..., g_nx_n - f_n, 1 - gy \rangle \subset \mathbb{K}[y, t_1, ..., t_m, x_1, ..., x_n]$, what is the significance of the term 1 - gy in the construction of $J$?
- It prevents the denominators $g_i$ from vanishing simultaneously. (correct)
- It simplifies the computation of Gröbner bases.
- It guarantees that the variety $V(J)$ is non-empty.
- It ensures that the variables $x_i$ are algebraically independent.
Suppose you have a rational parametrization $\varphi: (u,v) \mapsto (\frac{u^2}{v}, \frac{v^2}{u}, u)$. What is the first step in finding an implicit equation for the image of $\varphi$ using polynomial implicitization?
Suppose you have a rational parametrization $\varphi: (u,v) \mapsto (\frac{u^2}{v}, \frac{v^2}{u}, u)$. What is the first step in finding an implicit equation for the image of $\varphi$ using polynomial implicitization?
Let $I = \langle vx - u^2, uy - v^2, z - u \rangle \subset \mathbb{Z}[u, v, x, y, z]$. If $I_2 = I \cap \mathbb{Z}[x, y, z] = \langle x^2y - z^3 \rangle$, and $V(I_2) = V(x^2y - z^3) \cup V(z)$, why is $V(I_2)$ NOT the Zariski closure of the image of $\varphi$?
Let $I = \langle vx - u^2, uy - v^2, z - u \rangle \subset \mathbb{Z}[u, v, x, y, z]$. If $I_2 = I \cap \mathbb{Z}[x, y, z] = \langle x^2y - z^3 \rangle$, and $V(I_2) = V(x^2y - z^3) \cup V(z)$, why is $V(I_2)$ NOT the Zariski closure of the image of $\varphi$?
Given the setup for rational implicitization with the map $\varphi: \mathbb{K}^m / W \rightarrow \mathbb{K}^n$, what is the purpose of introducing the map $\iota: \mathbb{K}^m / W \rightarrow \mathbb{K}^{n+m}$?
Given the setup for rational implicitization with the map $\varphi: \mathbb{K}^m / W \rightarrow \mathbb{K}^n$, what is the purpose of introducing the map $\iota: \mathbb{K}^m / W \rightarrow \mathbb{K}^{n+m}$?
In the context of rational maps, suppose a coordinate $\sigma_{ij}$ is specified by a rational function of concentration parameters. What does this imply about the nature of the map itself?
In the context of rational maps, suppose a coordinate $\sigma_{ij}$ is specified by a rational function of concentration parameters. What does this imply about the nature of the map itself?
Consider the diagram involving the maps $\iota$, $\pi_m$, and $\varphi$ in the rational implicitization process. Which statement best describes the relationship between these maps?
Consider the diagram involving the maps $\iota$, $\pi_m$, and $\varphi$ in the rational implicitization process. Which statement best describes the relationship between these maps?
Why is it important to consider subfields $\mathbb{K}$ of $\mathbb{C}$ in the context of this proof, rather than just assuming $\mathbb{K} = \mathbb{C}$?
Why is it important to consider subfields $\mathbb{K}$ of $\mathbb{C}$ in the context of this proof, rather than just assuming $\mathbb{K} = \mathbb{C}$?
Given that $\mathbb{K}$ is a subfield of $\mathbb{C}$, why does this imply that $\mathbb{K}$ contains $\mathbb{Z}$ and $\mathbb{Q}$?
Given that $\mathbb{K}$ is a subfield of $\mathbb{C}$, why does this imply that $\mathbb{K}$ contains $\mathbb{Z}$ and $\mathbb{Q}$?
How does proving that $g_i \circ \varphi$ is the zero polynomial over $\mathbb{K}$ demonstrate that $V_{\mathbb{C}}(I_m) \subset Z_{\mathbb{C}}$?
How does proving that $g_i \circ \varphi$ is the zero polynomial over $\mathbb{K}$ demonstrate that $V_{\mathbb{C}}(I_m) \subset Z_{\mathbb{C}}$?
Why does the fact that $\mathbb{K}$ is infinite imply that if $g_i \circ \varphi(a) = 0$ for all $a \in \mathbb{K}^m$, then $g_i \circ \varphi$ is the zero polynomial?
Why does the fact that $\mathbb{K}$ is infinite imply that if $g_i \circ \varphi(a) = 0$ for all $a \in \mathbb{K}^m$, then $g_i \circ \varphi$ is the zero polynomial?
In the context of the proof, what role does the elimination ideal $I_m$ play?
In the context of the proof, what role does the elimination ideal $I_m$ play?
What is the significance of showing that $V_{\mathbb{K}}(I_m) \subset Z_{\mathbb{K}}$ in the context of proving that $V_{\mathbb{K}}(I_m)$ is the smallest variety in $\mathbb{K}^n$ containing $\varphi(\mathbb{K}^m)$?
What is the significance of showing that $V_{\mathbb{K}}(I_m) \subset Z_{\mathbb{K}}$ in the context of proving that $V_{\mathbb{K}}(I_m)$ is the smallest variety in $\mathbb{K}^n$ containing $\varphi(\mathbb{K}^m)$?
The proof considers $V = V(I) \subset \mathbb{K}^{n+m}$ as the graph of $\varphi$. How does this construction aid in proving the main result?
The proof considers $V = V(I) \subset \mathbb{K}^{n+m}$ as the graph of $\varphi$. How does this construction aid in proving the main result?
What is the significance of the step where the proof transitions from considering solutions in $\mathbb{C}^n$ to considering only solutions in $\mathbb{K}^n$?
What is the significance of the step where the proof transitions from considering solutions in $\mathbb{C}^n$ to considering only solutions in $\mathbb{K}^n$?
Consider a polynomial parametrization $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$ defined by $x_i = f_i(t_1, ..., t_m)$ for $i = 1, ..., n$. Which statement accurately describes the relationship between $\varphi(\mathbb{K}^m)$ and affine varieties?
Consider a polynomial parametrization $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$ defined by $x_i = f_i(t_1, ..., t_m)$ for $i = 1, ..., n$. Which statement accurately describes the relationship between $\varphi(\mathbb{K}^m)$ and affine varieties?
Let $V = V(x_1 - f_1, ..., x_n - f_n) \subset \mathbb{K}^{n+m}$, where $x_i = f_i(t_1, ..., t_m)$ defines a polynomial parametrization. What is the geometric interpretation of $V$?
Let $V = V(x_1 - f_1, ..., x_n - f_n) \subset \mathbb{K}^{n+m}$, where $x_i = f_i(t_1, ..., t_m)$ defines a polynomial parametrization. What is the geometric interpretation of $V$?
In the context of the Rational Implicitization Theorem, what role does the ideal $J = \langle g, x-f_1, ..., z-f_m, gy-1 \rangle$ play in relating the parametrization $\Psi : \mathbb{K}^m / W \to \mathbb{K}^n$ to its image?
In the context of the Rational Implicitization Theorem, what role does the ideal $J = \langle g, x-f_1, ..., z-f_m, gy-1 \rangle$ play in relating the parametrization $\Psi : \mathbb{K}^m / W \to \mathbb{K}^n$ to its image?
Consider a scenario where you are using elimination theory to solve an implicitization problem. If a component, $Z$, is unintentionally introduced during the parametrization process and is later removed by taking the Zariski closure, what does this removal signify?
Consider a scenario where you are using elimination theory to solve an implicitization problem. If a component, $Z$, is unintentionally introduced during the parametrization process and is later removed by taking the Zariski closure, what does this removal signify?
Given the maps $\iota: \mathbb{K}^m \rightarrow \mathbb{K}^{n+m}$ defined by $\iota(t_1, ..., t_m) = (t_1, ..., t_m, f_1(t_1, ..., t_m), ..., f_n(t_1, ..., t_m))$ and $\pi_m: \mathbb{K}^{n+m} \rightarrow \mathbb{K}^n$ defined by $\pi_m(t_1, ..., t_m, x_1, ..., x_n) = (x_1, ..., x_n)$, how are these maps related to the parametrization $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$?
Given the maps $\iota: \mathbb{K}^m \rightarrow \mathbb{K}^{n+m}$ defined by $\iota(t_1, ..., t_m) = (t_1, ..., t_m, f_1(t_1, ..., t_m), ..., f_n(t_1, ..., t_m))$ and $\pi_m: \mathbb{K}^{n+m} \rightarrow \mathbb{K}^n$ defined by $\pi_m(t_1, ..., t_m, x_1, ..., x_n) = (x_1, ..., x_n)$, how are these maps related to the parametrization $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$?
Suppose $V$ is an algebraic set and $g(T)$ is a polynomial such that the dimension of $g(V)$ is less than the dimension of $V$. What can be inferred about the relationship between $g(T)$ and $V$?
Suppose $V$ is an algebraic set and $g(T)$ is a polynomial such that the dimension of $g(V)$ is less than the dimension of $V$. What can be inferred about the relationship between $g(T)$ and $V$?
What does the Polynomial Implicitization Theorem state regarding the ideal $I = \langle x_1 - f_1, ..., x_n - f_n \rangle \subset \mathbb{K}[t_1, ..., t_m, x_1, ..., x_n]$ and the Zariski closure of $\varphi(\mathbb{K}^m)$ in $\mathbb{K}^n$, assuming $\mathbb{K}$ is an infinite field?
What does the Polynomial Implicitization Theorem state regarding the ideal $I = \langle x_1 - f_1, ..., x_n - f_n \rangle \subset \mathbb{K}[t_1, ..., t_m, x_1, ..., x_n]$ and the Zariski closure of $\varphi(\mathbb{K}^m)$ in $\mathbb{K}^n$, assuming $\mathbb{K}$ is an infinite field?
What is the primary role of semi-algebraic sets in the context of statistical models?
What is the primary role of semi-algebraic sets in the context of statistical models?
In the context of polynomial parametrization and implicitization, what is the significance of finding the smallest affine variety containing $\varphi(\mathbb{K}^m)$?
In the context of polynomial parametrization and implicitization, what is the significance of finding the smallest affine variety containing $\varphi(\mathbb{K}^m)$?
Consider the parametrization $\varphi: \mathbb{R} \rightarrow \mathbb{R}^2$ given by $\varphi(t) = (t^2, t^3)$. What is the implicit equation of the Zariski closure of the image of $\varphi$?
Consider the parametrization $\varphi: \mathbb{R} \rightarrow \mathbb{R}^2$ given by $\varphi(t) = (t^2, t^3)$. What is the implicit equation of the Zariski closure of the image of $\varphi$?
Why are exponential families considered important and commonly studied in statistical modeling?
Why are exponential families considered important and commonly studied in statistical modeling?
Given a rational parametrization $\Psi : \mathbb{K}^m / W \to \mathbb{K}^n$, where $\mathbb{K}$ is an infinite field, what does the notation $\mathbb{K}^m / W$ represent?
Given a rational parametrization $\Psi : \mathbb{K}^m / W \to \mathbb{K}^n$, where $\mathbb{K}$ is an infinite field, what does the notation $\mathbb{K}^m / W$ represent?
Let $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$ be a polynomial parametrization. If $I = \langle x_1 - f_1, ..., x_n - f_n \rangle \subset \mathbb{K}[t_1, ..., t_m, x_1, ..., x_n]$, how is the elimination ideal $I_m = I \cap \mathbb{K}[x_1, ..., x_n]$ related to the implicitization problem?
Let $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$ be a polynomial parametrization. If $I = \langle x_1 - f_1, ..., x_n - f_n \rangle \subset \mathbb{K}[t_1, ..., t_m, x_1, ..., x_n]$, how is the elimination ideal $I_m = I \cap \mathbb{K}[x_1, ..., x_n]$ related to the implicitization problem?
Suppose you have a polynomial parametrization $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$. What are the general steps one would take to find an implicit representation of the image of $\varphi$?
Suppose you have a polynomial parametrization $\varphi: \mathbb{K}^m \rightarrow \mathbb{K}^n$. What are the general steps one would take to find an implicit representation of the image of $\varphi$?
In the context of algebraic statistics, what does it mean to extract 'meaningful' statistical models by defining statistical models algebraically?
In the context of algebraic statistics, what does it mean to extract 'meaningful' statistical models by defining statistical models algebraically?
What distinguishes semi-algebraic statistical models from traditional algebraic models in statistics?
What distinguishes semi-algebraic statistical models from traditional algebraic models in statistics?
Given the parametrization $x_i = f_i(t_1, ..., t_m)$ for $i = 1, ..., n$, what is the initial step in the algorithm for solving the proliferation problem for polynomial parametrizations?
Given the parametrization $x_i = f_i(t_1, ..., t_m)$ for $i = 1, ..., n$, what is the initial step in the algorithm for solving the proliferation problem for polynomial parametrizations?
In the example parametrization $\varphi : t \mapsto ((1-t)^2, 2t(1-t), t^2)$, what is the significance of computing the Grobner basis $G = {p_1^2 - 4p_0p_2, p_0 + p_1 + p_2 - 1}$?
In the example parametrization $\varphi : t \mapsto ((1-t)^2, 2t(1-t), t^2)$, what is the significance of computing the Grobner basis $G = {p_1^2 - 4p_0p_2, p_0 + p_1 + p_2 - 1}$?
For the concentration matrix $K$ in the context of concentration models, which of the following statements is correct?
For the concentration matrix $K$ in the context of concentration models, which of the following statements is correct?
What is the significance of computing the intersection $G \cap \mathbb{Z}[x_1, ..., x_n]$ in the algorithm for solving the proliferation problem?
What is the significance of computing the intersection $G \cap \mathbb{Z}[x_1, ..., x_n]$ in the algorithm for solving the proliferation problem?
Given $\overline{\varphi(\mathbb{R})} = V(p_1^2 - 4p_0p_2, p_0 + p_1 + p_2 - 1)$, which of the following describes the set $M = \overline{\varphi((0,1))}$?
Given $\overline{\varphi(\mathbb{R})} = V(p_1^2 - 4p_0p_2, p_0 + p_1 + p_2 - 1)$, which of the following describes the set $M = \overline{\varphi((0,1))}$?
In the context of linear concentration models, given $\Xi = { a\begin{bmatrix} 1 & a & 1 \ a & 1 & a \ 1 & a & 1 \end{bmatrix}+ b \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}:a, b \in \mathbb{R}} \subset S^3$, what does the set $M = {\varepsilon \sim N(0, \varepsilon) :\Sigma^{-1} \in \Xi \cap PD_3}$ represent?
In the context of linear concentration models, given $\Xi = { a\begin{bmatrix} 1 & a & 1 \ a & 1 & a \ 1 & a & 1 \end{bmatrix}+ b \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}:a, b \in \mathbb{R}} \subset S^3$, what does the set $M = {\varepsilon \sim N(0, \varepsilon) :\Sigma^{-1} \in \Xi \cap PD_3}$ represent?
Given $\varphi(K) = {\sigma_{ij} = (-1)^{i+j}\tfrac{|K^{[3] \setminus {j}}, [3] \setminus {i}}|}{|K|}: i,j \in [3]}$, why is it important to extend theorems to cover models with a rational parametrization like $\varphi$?
Given $\varphi(K) = {\sigma_{ij} = (-1)^{i+j}\tfrac{|K^{[3] \setminus {j}}, [3] \setminus {i}}|}{|K|}: i,j \in [3]}$, why is it important to extend theorems to cover models with a rational parametrization like $\varphi$?
What is the purpose of computing a Grobner basis with a lexicographic term order where every $t_j$ is greater than every $x_i$?
What is the purpose of computing a Grobner basis with a lexicographic term order where every $t_j$ is greater than every $x_i$?
Flashcards
Polynomial Parametrization
Polynomial Parametrization
A way to represent variables x₁, ..., xₙ using polynomial functions f₁, ..., fₙ of parameters t₁, ..., tₘ.
Parametrization Function ((\varphi))
Parametrization Function ((\varphi))
A function (\varphi) that maps points from (\mathbb{K}^m) to (\mathbb{K}^n) using polynomial functions.
Implicitization Problem
Implicitization Problem
Finding the smallest affine variety that contains the image of a polynomial parametrization.
Variety V
Variety V
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V as Graph of (\varphi)
V as Graph of (\varphi)
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Embedding Function ((\iota))
Embedding Function ((\iota))
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Projection Function ((\pi_m))
Projection Function ((\pi_m))
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Polynomial Implicitization Theorem
Polynomial Implicitization Theorem
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(\overline{\varphi(\mathbb{K}^m)})
(\overline{\varphi(\mathbb{K}^m)})
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V
V
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(\pi_m(V))
(\pi_m(V))
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(V_{\mathbb{Z}}(I_m))
(V_{\mathbb{Z}}(I_m))
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(\mathbb{C})
(\mathbb{C})
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(\mathbb{K})
(\mathbb{K})
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(\varphi)
(\varphi)
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(I_m)
(I_m)
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Proliferation Problem Algorithm
Proliferation Problem Algorithm
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Groebner Basis
Groebner Basis
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Zariski Closure
Zariski Closure
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Variety V(S)
Variety V(S)
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Linear Concentration Model
Linear Concentration Model
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Concentration Matrix
Concentration Matrix
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Positive Definite Matrix
Positive Definite Matrix
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$\Delta_0^2$
$\Delta_0^2$
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Rational Map
Rational Map
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Rational Parametrization
Rational Parametrization
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Rational Implicitization Problem
Rational Implicitization Problem
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Rational Parametrization Function
Rational Parametrization Function
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The Set W
The Set W
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Extended Embedding Function ((\iota))
Extended Embedding Function ((\iota))
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The Ideal J
The Ideal J
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Projection Function ((\pi_{m+1}))
Projection Function ((\pi_{m+1}))
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Elimination Variety
Elimination Variety
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Rational Parametrization (\Psi)
Rational Parametrization (\Psi)
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Rational Implicitization goal
Rational Implicitization goal
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Exponential Families
Exponential Families
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Algebraic Statistical Models
Algebraic Statistical Models
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Semialgebraic Statistical Models
Semialgebraic Statistical Models
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Computational Tools
Computational Tools
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Field (\mathbb{K})
Field (\mathbb{K})
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Study Notes
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Parametrization of x, starts with a polynomial, where x is equal to a function f of t
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f₁, ..., fₙ ∈ k[t₁, ..., tₘ], where k is a field
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Geometric representation is a function of phi, mapping k^m to k^n
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φ: (t₁, ..., tₘ) ↦ (f₁(t₁, ..., tₘ), ..., fₙ(t₁, ..., tₘ))
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φ(k^m) ⊆ k^n is a subset of k^n, parametrized by equations
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φ(k^m) does not need to be an affine variety, hence the need to find the smallest affine variety containing φ(k^m)
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The implicitization problem seeks to find polynomials F₁, ..., Fₛ such that V(F₁, ..., Fₛ) is the smallest affine variety containing φ(k^m)
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Equations define a variety: V = V(x₁ - f₁, ..., xₙ - fₙ) ⊆ k^(n+m)
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Implicitization relates to elimination
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Points in V can be written as (t₁, ..., tₘ, f₁(t₁, ..., tₘ), ..., fₙ(t₁, ..., tₘ))
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V represents the graph of φ
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φ: ℝ → ℝ (t ↦ t² = x), e.g., (t, x) ∈ V(ℝ-x) is (t, t²) ∈ ℝ
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Auxiliary functions: ι: k^m → k^(n+m) (t₁, ..., tₘ ↦ (t₁, ..., tₘ, f₁(t₁, ..., tₘ), ..., fₙ(t₁, ..., tₘ)))
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πₘ: k^(n+m) → k^n (t₁, ..., tₘ, x₁, ..., xₙ ↦ (x₁, ..., xₙ))
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Diagram: k^m -> k^(n+m) -> k^n
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φ = πₘ o ι
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ι(k^m) = V.
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φ(k^m) = πₘ(ι(k^m)) = πₘ(V)
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The image of the parametrization equals the projection of its graph
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Elimination and closure theorems solve the implicitization problem
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The Polynomial Implicitization Theorem states that for an infinite field k and polynomial parametrization φ: k^m → k^n, define the ideal I = <x₁ - f₁, ..., xₙ - fₙ> ⊂ k[t₁, ..., tₘ, x₁, ..., xₙ]. Then, V(Iₘ) is the Zariski closure of φ(k^m) in k^n
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V(I(φ(k^m))) is the smallest affine variety containing φ(k^m)
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The goal is to show that V(Iₘ) is the smallest affine variety containing φ(k^m) by starting with V = V(I) ⊂ k^(n+m) as the graph of φ
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Assuming k = C and using the Closure Theorem finds that V(Iₘ) is the smallest affine variety containing φ(k^m), thus completing this part of the proof
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For k = C, let k be a subfield of C that forms a field under the same multiplication and addition
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k contains Z (integers) and Q (rationals), so k is infinite
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k could be non-algebraically closed, preventing direct application of the closure theorem.
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Vₖ(Iₘ) is the variety in k^n, and V_C(Iₘ) is the variety in C^n
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Expanding to a larger field does not alter the elimination ideal Iₘ, since the algorithm for computing Iₘ is unaffected by changing from k to C
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Need to prove that Vₖ(Iₘ) is the smallest variety in k^n containing φ(k^m)
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φ(k^m) = πₘ(V) ⊆ Vₖ(Iₘ)
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Lemma states V = V(x₁ - f₁, ..., xₙ - fₙ) over k^(n+m)
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Let Zₖ = Vₖ(g₁, ..., gₛ) ⊆ k^n be a variety in k^n such that φ(k^m) ⊆ Zₖ
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Since gᵢ(a) = 0 ∀ a ∈ Zₖ then gᵢ(a) = 0 ∀ a ∈ φ(k^m) meaning gᵢ o φ vanishes on all of φ(k^m) and gᵢ o φ ∈ k[t₁, ..., tₘ]
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gᵢ o φ(a) = 0 ∀ a ∈ k^m, and since k is infinite, gᵢ o φ is a 0-polynomial ∀i
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Means gᵢ vanishes on all of φ(k^m), so Zₖ = Vₖ(g₁, ..., gₛ) in k^n containing φ(k^m)
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V_C(Iₘ) ⊆ Zₖ in C^n (by case k=C)
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By considering only solutions in k^n, it follows that Vₖ(Iₘ) ⊆ Zₖ, proving the result for k ⊆ C
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If k is not a subfield of C, one can prove there is an algebraically closed field containing k and use similar arguments
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The algorithm for solving the parametrization problem for polynomial parametrizations uses the given parametrization xᵢ = fᵢ(t₁, ..., tₘ) ∀ i = 1, ..., n, with fᵢ ∈ k[t₁, ..., tₘ]
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Define the ideal I = <x₁ - f₁, ..., xₙ - fₙ> ⊂ k[t₁, ..., tₘ, x₁, ..., xₙ]
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Compute a Gröbner basis G for I with respect to lexicographic term order where every tᵢ is greater than every xᵢ
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Then take G ∩ k[x₁, ..., xₙ]
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Example case: φ: ℝ → ℝ³ via φ(t) = ((b-t)², 2t(1-t), t²)
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Compute I = <p₀ - (1-t)², p₁ - 2t(1-t), p₂ - t²>
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Gröbner basis is G = {p₁² + 4p₀p₂ + 4p₂² - 4p₂, p₀ + p₁ + p₂ - 1, 2t - p₁ - 2p₂}
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φ(ℝ) = V(p₁² + 4p₀p₂ + 4p₂² - 4p₂, p₀ + p₁ + p₂ - 1)
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<p₁² + 4p₀p₂ + 4p₂² - 4p₂, p₀ + p₁ + p₂ - 1> = <p₁² - 4p₀p₂, p₀ + p₁ + p₂ - 1>
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φ(ℝ) = V(p₁² - 4p₀p₂, p₀ + p₁ + p₂ - 1)
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M = ρ({0,1}) = φ(ℝ) = V(p₁² - 4p₀p₂, p₀ + p₁ + p₂ - 1)
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Semialgebraic description of the binomial model for n=2 is M = V(p₁² - 4p₀p₂) ∩ Δ²₀
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The example of concentration models, where Z = [Z₁, Z₂, Z₃]ᵀ ~ N(Ω, Σ) and Σ⁻¹ = K (concentration matrix)
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K is a 3x3 real symmetric matrices, and a 3x3 positive definite matrices
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Need to extend theorem to cover models with a rational parametrization
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The map is a rational map because each σᵢⱼ coordinate is specified by a rational function of the concentration parameters (quotient of polynomials)
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Statistical models require extending solutions of the implicitization problem to rational parametrizations
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Rational parametrizations: φ: ℝ² → ℝ³ where φ:(u, v) ↦ (u²/v, v²/u, u) = (x, y, z)
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Notice that if (x, y, z) ∈ Im(φ), then x²y - z³ = 0
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Polynomial implicitization applies by clearing denominators, resulting in ideal I = <vx - u², uy - v², z - u> ⊂ k[u, v, x, y, z]
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Exercise: I₂ = I ∩ k[x, y, z] = <z(x²y - z³)>, so V(I₂) = V(x²y - z³) ∪ V(z)
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V(I₂) is not a Zariski closure since Im(φ) ⊂ V(x²y - z²)
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Need to remove V(z)
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A general setup for xᵢ = fᵢ(t₁, ..., tₘ)/gᵢ(t₁, ..., tₘ), with fᵢ, gᵢ ∈ k[t₁, ..., tₘ]
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The map cannot be defined on all of k^m since denominators cannot be zero
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φ: k^m \ W → k^n defined as φ: (t₁, ..., tₘ) ↦ (f₁(t₁, ..., tₘ)/g₁(t₁, ..., tₘ), ..., fₙ(t₁, ..., tₘ)/gₙ(t₁, ..., tₘ)), where W = V(g₁g₂...gₙ) ⊂ k^m
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The Rational Implicitization Problem finds the smallest variety of k^n containing φ(k^m \ W)
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Adapting maps yields ι: k^m \ W → k^(n+m), where I = <x₁g₁ - f₁, ..., xₙgₙ - fₙ>
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Ideal from clearing denominators
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Adding an extra dimension avoids vanishing denominators
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Define g = g₁...gₙ, so J = <g₁x₁ - f₁, ..., gₙxₙ - fₙ, 1 - gy> ⊂ k[y, t₁, ..., tₘ, x₁, ..., xₙ]
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(1 - gy) ensures that denominators do not vanish on V(J)!
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2^(n+m+1), creating maps j and π
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π: 2^m \ W → 2^(n+m+1)
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π:( 2^m \ W ) → (1/g(m),tytm, f₁/g , fₙ/g)
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φ= π ∘ i
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Claim that j ( W) = V(I)
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J ( W)CV(I) by definitions.
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If (W) =V(J) Conversely, if (yW, x)
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(1 - gy) ensures that each variable remains defined by its respective expression
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Since the function y = 1/g( W ), our point is in j ( W ) Imp: (J W) is projection of V(I) is an example.
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Theorem (Rational Implicitization): Let k be an infinite field, with rational parametrization as above and J = <gx - f, gy-1> Z[y, t, x] where s= gg. Then for V(J) m,w is the smallest affine variety containing
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To compute the analogy of the algorithm for solving the implicitization problem as for polynomial parametrizations.
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Continuing our small example is useful, with the new variable becomes, J = <z-, 1- wy> k[u,v,x,,у,z]
m z]] =
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A subtlety worth considering statistical models has important information
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4 is a rational map and Zm
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Want to note that which leads to V ( I(M))
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