Merge pull request #6919 from albert-github/feature/bug_Polynomial_formula_2

Polynomial: replace images with formulas

Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--PolynomialSubresultants.h

@@ -11,8 +11,17 @@ Let
 is the outermost variable.
 The \f$ i\f$-th subresultant (with \f$ i=0,\ldots,\min\{n,m\}\f$) is defined by
 
-\image html subresultant_def.png
-\image latex subresultant_def.png
+\f[
+\mathrm{Sres}_{i}(p,q) = \det
+\begin{pmatrix}
+    p_{n} & \dots & & \dots & p_{2i-m+2} & x^{m-i-1}p  \\
+    & \ddots & & & \vdots & \vdots\\
+    & & p_{n} & \dots & p_{i+1} & p \\
+    q_{m} & \dots & & \dots & q_{2i-n+2} & x^{n-i-1}q  \\
+    & \ddots & & & \vdots & \vdots\\
+    & & q_{m} & \dots & q_{i+1} & q
+\end{pmatrix}
+\f]
 
 where \f$ p_i\f$ and \f$ q_i\f$ are set to zero if \f$ i<0\f$.
 In the case that \f$ n=m\f$, \f$ \mathrm{Sres_n}\f$ is set to \f$ q\f$.

Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--Resultant.h

@@ -17,8 +17,18 @@ where
 \f[ g = g_nx^n + \dots + g_0. \f]
 The resultant of \f$ f\f$ and \f$ g\f$ is defined as the determinant of the <I>Sylvester matrix</I>:
 
-\image html sylvester_matrix.png
-\image latex sylvester_matrix.png
+\f[
+\begin{pmatrix}
+    f_{m} & \dots & f_{0}  \\
+    & f_{m} & \dots & f_{0}  \\
+    & & \ddots & & \ddots  \\
+    & & & f_{m} & \dots & f_{0}  \\
+    g_{n} & \dots & g_{0}  \\
+    & g_{n} & \dots & g_{0}  \\
+    & & \ddots & & \ddots  \\
+    & & & g_{n} & \dots & g_{0}
+\end{pmatrix}
+\f]
 
 Note that this is a \f$ (n+m)\times(n+m)\f$ matrix as there are \f$ n\f$ rows for \f$ f\f$
 and \f$ m\f$ rows that are used for \f$ g\f$. The blank spaces are supposed to be

Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--SturmHabichtSequence.h

@@ -15,8 +15,15 @@ Let \f$ n:=\deg f\f$ and \f$ \delta_k:=(-1)^{k(k+1)/2}\f$.
 For \f$ k\in\{0,\ldots,n\}\f$, the <I>\f$ k\f$-th Sturm-Habicht polynomial</I>
 of \f$ f\f$ is defined as:
 
-\image html sturm_habicht_def.png
-\image latex sturm_habicht_def.png
+\f[
+\mathrm{Stha}_{k}(f) = \left \{
+\begin{array}{cll}
+f & \text{if} & k = n \\
+f' & \text{if} & k = n - 1 \\
+\delta_{n - k - 1}\mathrm{Sres}_{k}(f,f') & \text{if} & 0 \leq k \leq n - 2
+\end{array}
+\right .
+\f]
 
 where \f$ \mathrm{Sres}_k(f,f')\f$ is defined
 as in the concept `PolynomialTraits_d::PolynomialSubresultants`.