diff --git a/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--PolynomialSubresultants.h b/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--PolynomialSubresultants.h index 85d45c187ce..3418ef03626 100644 --- a/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--PolynomialSubresultants.h +++ b/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$. diff --git a/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--Resultant.h b/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--Resultant.h index 5df1182a4b0..e337487c90d 100644 --- a/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--Resultant.h +++ b/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--Resultant.h @@ -17,8 +17,18 @@ where \f[ g = g_nx^n + \dots + g_0. \f] The resultant of \f$ f\f$ and \f$ g\f$ is defined as the determinant of the Sylvester matrix: -\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$. The blank spaces are supposed to be diff --git a/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--SturmHabichtSequence.h b/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--SturmHabichtSequence.h index 5390c1b9e5d..c314610f9ee 100644 --- a/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--SturmHabichtSequence.h +++ b/Polynomial/doc/Polynomial/Concepts/PolynomialTraits_d--SturmHabichtSequence.h @@ -15,8 +15,15 @@ Let \f$ n:=\deg f\f$ and \f$ \delta_k:=(-1)^{k(k+1)/2}\f$. For \f$ k\in\{0,\ldots,n\}\f$, the \f$ k\f$-th Sturm-Habicht polynomial of \f$ 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`. diff --git a/Polynomial/doc/Polynomial/fig/sturm_habicht_def.png b/Polynomial/doc/Polynomial/fig/sturm_habicht_def.png deleted file mode 100644 index a0aace84a66..00000000000 Binary files a/Polynomial/doc/Polynomial/fig/sturm_habicht_def.png and /dev/null differ diff --git a/Polynomial/doc/Polynomial/fig/subresultant_def.png b/Polynomial/doc/Polynomial/fig/subresultant_def.png deleted file mode 100644 index 88309188129..00000000000 Binary files a/Polynomial/doc/Polynomial/fig/subresultant_def.png and /dev/null differ diff --git a/Polynomial/doc/Polynomial/fig/sylvester_matrix.png b/Polynomial/doc/Polynomial/fig/sylvester_matrix.png deleted file mode 100644 index 30b13c4f1b3..00000000000 Binary files a/Polynomial/doc/Polynomial/fig/sylvester_matrix.png and /dev/null differ