polished up to specialized algotithms, just missing polishing that section

Author: fragandi

source/ch-invarianttheory.ptx

@@ -3,7 +3,7 @@
     <title>Invariant Theory</title>
     <introduction>
         <p>
-This chapter is co-authored by Francesca Gandini, Sumner Strom, Al Ashir Intisar. 
+This chapter is co-authored by Francesca Gandini, Al Ashir Intisar, and Sumner Strom. 
 In this chapter we will present an overview of the theory behind the algorithms implemented in the <url href="https://www.macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/InvariantRing/html/index.html">InvariantRing</url> software package 
 in the open-source Computer Algebra System <url href="http://www2.macaulay2.com">Macaulay2 (M2)</url>.
         </p>
@@ -24,7 +24,7 @@ in the open-source Computer Algebra System <url href="http://www2.macaulay2.com"
         <p>
           We finish the chapter with a selection of examples that illustrate the current capabilities of the InvariantRing package.
           You can run the provided code in your local installation of M2 or go to the online version and execute the code cells on your browser.
-          This works well even on mobile devices!
+           This works  well even on mobile devices!
         </p>
     </introduction>
 
@@ -72,7 +72,7 @@ x \\
         In general, most of the results in this chapter hold in the non-modular case 
         i.e., when the characteristics of the field does not divide the order of the group. 
         As of now finite fields are not fully supported by the current version of the InvariantRing package 
-        and such functionalities is an active area of development.  
+        and such functionalities is an active area of development.   
 </p>
 
 <p>
@@ -203,10 +203,10 @@ and we can check that <m>R_G(x+y)=x</m> is indeed invariant.
         <title>Hilbert Ideal</title>
         <p>
                 To describe a more sophisticated approach to the search for minimal algebra generators for an invariant ring, we will actually need to consider an ideal instead!
-  Note: for any <m>\{ f_1,..., f_s\} \subseteq R</m>, the ideal generatd by <m>\{f_1,...f_s\}</m> 
+  Note: for  any <m>\{ f_1,..., f_s\} \subseteq R</m>, the ideal generatd by <m>\{f_1,...f_s\}</m> 
   and the subalgebra generated by  <m>\{f_1,...f_s\}</m> over <m>\mathbb{K}</m> are very different objects.
        </p>
-       <definition xml:id="def-">
+       <definition>
           <p>
             Let <m>J_G := R(R^G)_+</m> be the ideal in <m>R</m> generated by all positive degree invariants. 
             We call <m>J_G</m> the Hilbert Ideal for this action of <m>G</m>.
@@ -222,77 +222,118 @@ and we can check that <m>R_G(x+y)=x</m> is indeed invariant.
        then you can apply the Reynolds operator <m>R^G</m> to obtain a new generator that is. You can now replace the old generator 
        with this new one and still get the same ideal. What is special here is that a set of ideal generators work as algebra generators!
        Computationally, algebra generators are much harder to find as there is no guarantee to have finitely many of them. 
-       However, Hilbert Basis Theorem tells us that every ideal is finitely generated.
-          </p>
+       However, the Hilbert Basis Theorem tells us that every ideal is finitely generated.
+      </p>
     </subsection>
+
     <subsection xml:id="subsec-presentations">
         <title>Presentations</title>
-            <p>
-                <definition><p>Definition: Let <m>S = \mathbb{K}[f_1,...f_s] \subset R</m>.
-                A presentation of <m>S</m> is a map, <me>T=: \mathbb{K}[u_1,...u_s] \xrightarrow{\phi}S</me>
-                such that <m>\frac{T}{\text{ker}(\phi)} \cong S</m> With the syzygies of <m>f_i</m>'s
-                giving the presentation ideal.
-                </p></definition>
+            <p> When we say that  <m>\{_1,...f_s\} \subseteq R</m> are minimal generators for a subring <m>S</m>, 
+              we do not exclude the possibility that there is some relation, some polynomial identity, that they satisfy 
+              as elements in the bigger ring <m>R</m>. We can describe the relations between the generators via a <term>presentation</term>
+              of the subring. 
             </p>
-            <p>
+
+                <definition>
+                  <p> Let <m>S = \mathbb{K}[f_1,...f_s] \subset R</m>.
+                A presentation of <m>S</m> is a map, <me>T=: \mathbb{K}[u_1,...u_s] \xrightarrow{\phi}S</me>
+                such that <m>\frac{T}{\text{ker}(\phi)} \cong S</m>. We call the elements of the presentation ideal <m>\text{ker}(\phi)</m> 
+               the syzygies of <m>f_i</m>'s.
+                </p>
+              </definition>
+            
+              <p>
+                 Algorithms for finding generators for ideals have been intensely studied and especially in relation with the theory of Groebener bases.
+                 We cannot go in the details of these tools, but what is of notice is that they are implemented in Macaulay2 
+                 and so we can rely on them in our implementation. In particular, these methods are particularly effective in solving problems 
+                 in Elimnination Theory.  Often the goal is to compute an ideal of relations hoping that this is less complicated than the original structure, 
+                 possibly elimnating some variables.
+              </p>
+           
                 <proposition><p>(Elimination Theory): In <m>S \bigotimes \mathbb{K}[u_1,...,u_s] = \mathbb{K}[x_1,...,x_n,u_1,...u_s]</m> consider the ideal,
                 <me>I = (u_i - f_x(\bar x) | \, \left\langle f_i\right\rangle = S</me>
                 Then,
                 <me>\text{ker} (\phi)= I \cap \mathbb{K}[u_1,...,u_s]</me>
                 </p></proposition>
-            </p>
-            <p>
+           
                 <algorithm><p> Compute a Groebner Basis <m>G</m> for <m>I</m> with elimination order for the <m>x</m>'s.
                 Then, <m>G \cap \mathbb{K}[y_1,...y_s]</m>  is the Groebner Basis for <m>ker \phi</m>
                 </p></algorithm>
-            </p>
+           
     </subsection>
+
     <subsection xml:id="subsec-graph-of-linear-actions">
         <title>Graph of Linear Actions</title>
-        <p>
-            <definition> <p>Let <m>G \leq GL_n(\mathbb{K}), \,\, G\curvearrowright \mathbb{K}^n =:V, \,\, |G|\infty</m>.
+        <p> We can use Elimination Theory to solve our original problem of finding minimal generators for the ring of invariants.
+          We first need to construct a geometric description of the action of a group <m>G</m>.
+          </p>
+            <definition> <p>Let <m>G</m> be a finite matrix group in <m>GL_n(\mathbb{K})</m>.
             For <m>A\in G</m> consider,
-            <me>V_A = \{(\bar v, A\bar v)|\,\,v\in V\} \subseteq V\bigotimes V</me>
+            <me> V_A = \left\{ ( \bar v, A \bar v) \mid ,v \in V \right \} \subseteq V \bigoplus V </me>
             Then <m>A_G = \cup_{A\in G}V_A</m> is the subspace arrangement associated to the action of G.
             </p></definition>
-        </p>
+        
         <p>
-            Note: <m>V_A</m> is a linear subspace, <m>\mathbb{I}(V_A):=</m> set of polynomials vanishing on <m>\mathbb{V}_A</m> is a linear ideal.
-            Example: <me>V_{\begin{pmatrix}
+            Note that <m>V_A</m> is a linear subspace. 
+            So <m>\mathbb{I}(V_A)</m>, the set of polynomials vanishing on <m>V_A</m>, is an ideal 
+            generated by linear polynomials, we call this a <term>linear ideal</term>.
+  </p>
+
+  <example>
+    <p>
+     Consider <me>V_{\begin{pmatrix}
             1 \amp 0 \\
             0 \amp -1 \\
-            \end{pmatrix}} = \{(x_1,x_2,x_1,-x_2)\} = V(y_1,-x_1, y_2+x_2)</me>
-        </p>
+            \end{pmatrix}} = \{(x_1,x_2,x_1,-x_2) \mid x_1, x_2 \in V\} = \mathbb{V}(y_1-x_1, y_2+x_2)</me>
+      
+    </p>
+  </example>
+           
     </subsection>
     <subsection xml:id="subsec-subspace-arrangement-approach">
         <title>Subspace Arrangement Approach</title>
         <p>
+          The finite union of the subspaces <m>V_A</m>, denoted <m>\mathcal{A}</m> is a subspace arrangement, called the <term>group action arrangement</term>. 
+          Via Elimination Theory, we can use the vanishing ideal of <m>\mathcal{A}</m> to recover the Hilbert Ideal.
+        </p> 
+    
             <theorem><p>
-                (Dekseu): Let <m>I_G = \mathbb{I}(A_G) = \cap_{A\in G}\mathbb{I}(V_A) \subseteq \mathbb{K}[x_1,...x_n,y_1,...y_n].</m>
-            Then <me>(I_G +(y_1,...,y_n)) \cap \R = J_G</me> This uses elimination theory and the Hilbert ideal.
+                (Derksen): Let <m>I_G = \mathbb{I}(A_G) = \cap_{A\in G}\mathbb{I}(V_A) \subseteq \mathbb{K}[x_1,...x_n,y_1,...y_n].</m>
+            Then <me>(I_G +(y_1,...,y_n)) \cap \R = J_G.</me>
             </p> </theorem>
-        </p>
+        
         <p>
-            Note: The same approach works in the exterior algebra!
+           Recent work has shown that the same approach works in the exterior algebra.
         </p>
         <p>
-            <theorem><p>Let <m>I_G^{'} = \cap_{A\in G} \mathbb{I}(V_A) \subseteq \Lambda(\bar x, \bar y)</m>.
+            <theorem><p>(Gandini) Let <m>I_G^{'} = \cap_{A\in G} \mathbb{I}(V_A) \subseteq \Lambda(\bar x, \bar y)</m>.
             Then <me>(I_G^{'} +(y_1,...y_n)) \cap \Lambda(x_1,...,x_n) = J_G^{'} : = \Lambda(\bar x)(\Lambda(\bar x)^G)_+</me>
             </p></theorem>
         </p>
         <p>
-            Note: This approach is slow for polynomials, but might be fast for skew polynomials.
+           The exterior algebra approach has computational potential. 
+           Whilst Derken's approach leads to an algorithm with a slow implementation, 
+           there is some potential for a fast algorithm to be implemented for skew polynomials.
+           We aim to pursue this line of inquiry in the near future.
         </p>
     </subsection>
+  </section>
+  <section xml:id="sec-">
+    <title>Specialized algorithms</title>
+    <p>
+      For some specific types of actions we have faster and simpler algorithms to find invariants. 
+    </p>
     <subsection xml:id="subsec-AGWM">
-        <title>Abelian GPS and Weight Matrices</title>
+        <title>Abelian Groups and Weight Matrices</title>
         <p>
-            Let <m>G \cong \mathbb{Z}_d, \bigoplus....\bigoplus \mathbb{Z}_{dr}, \,\,\,\,\, d_i|d_{i+1}</m> for <m>1 \leq i \leq r-1</m>
-            <me>\left\langle g_1\right\rangle \bigoplus...\bigoplus\left\langle g_r \right\rangle, \,\,\,\,\, |g_i| =d_i</me>
+            Every abelian group <m>G</m> can be written in its invariant factos decomposition as 
+            <me>G \cong \mathbb{Z}_d, \bigoplus....\bigoplus \mathbb{Z}_{dr}, \,\,\,\,\, d_i|d_{i+1}, \, \, 1 \leq i \leq r-1</me>
+            where
+              <me>\left\langle g_1\right\rangle \bigoplus...\bigoplus\left\langle g_r \right\rangle, \,\,\,\,\, |g_i| =d_i</me>
             A diagonal action of <m>G</m> on <m>R</m> is given by
             <me>g_i \cdot x_j = \mu_i^{\omega ij}x_j</me>
-            for <m> \mu_i : d_i^{th}</m> primitive root of unity and <m>i \in [x]</m>,<m>j \in [n]</m>.
-             And encoded in the weight matrix <m>W = (\omega_{ij})_{ij} =  
+            for <m> \mu_i </m> a   <m>d_i^{th}</m> primitive root of unity, <m>i \in [x]</m>,<m>j \in [n]</m>.
+            We can encod the action in the weight matrix <m>W = (\omega_{ij})_{ij} =  
             \begin{pmatrix}
                 x_1 \amp \cdots     \amp   x_n   \\
                 \vdots \amp \ddots \amp      \\