A fast algorithm for the construction of integrity bases associated to symmetry-adapted polynomial representations: application to tetrahedral $\mathrm {XY_4}$ molecules

Abstract

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques, for the construction of symmetry-adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter polynomials. The present article extends its use to the construction of polynomial bases which span possibly, non-totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry-adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on $\mathrm {XY_4}$ type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper.

Appendices

Appendix 1: Generating functions and corresponding integrity bases for irreducible representations of $\mathrm {T}_d$

The $\mathrm {T}_d$ point group has five irreps: $A_1$, $A_2$, $E$, $F_1$ and $F_2$. The irrep $E$ is doubly degenerate, while the $F_1$ and $F_2$ irreps are triply degenerate. The procedure detailed in Sect. 2 is based on the knowledge of the generating functions $M^{\mathrm {T}_d}\left( \Gamma ^{\mathrm{final}};\Gamma ^{\mathrm{initial}};t\right)$, where $\Gamma ^{\mathrm{initial}}$ and $\Gamma ^{\mathrm{final}}$ are irreps of the group $\mathrm {T}_d$. The coefficient $c_n$ in the Taylor expansion $c_0\,+\,c_1t\,+\,c_2t^2+\cdots$ of the generating function gives the number of linearly independent $\Gamma ^{\mathrm{final}}$-covariant polynomials of degree $n$ that can be constructed from the objects in the initial $\Gamma ^{\mathrm{initial}}$ representation.

Each generating function $M^{\mathrm {T}_d}\left( \Gamma ^{\mathrm{final}};\Gamma ^{\mathrm{initial}};t\right)$ is the ratio of a numerator $\mathcal {N}\left( \Gamma ^{\mathrm{final}};\Gamma ^{\mathrm{initial}};t\right)$ over a denominator $\mathcal {D}\left( \Gamma ^{\mathrm{initial}};t\right)$:

$$\begin{aligned} M^{\mathrm {T}_d}\left( \Gamma ^{\mathrm{final}} ;\Gamma ^{\mathrm{initial}} ; t \right) = \frac{\mathcal {N}\left( \Gamma ^{\mathrm{final}};\Gamma ^{\mathrm{initial}};t\right) }{\mathcal {D}\left( \Gamma ^{\mathrm{initial}};t\right) } = \frac{\sum \nolimits _{k=1}^{N} t^{\nu _k}}{\prod \nolimits _{k=1}^{D} \left( 1-t^{\delta _k}\right) }, \end{aligned}$$

(39)

with $\nu _k \in \mathbb {N}$ and $\delta _k \in \mathbb {N}\backslash {}\left\{ 0\right\}$. The polynomial associated to a $\left( 1-t^{\delta _k}\right)$ term in the denominator is an invariant called a denominator polynomial of degree $\delta _k$ and is noted $I^{\left( \delta _k\right) }\left( \Gamma ^{\mathrm{initial}}\right)$. The polynomial associated to a $t^{\nu _k}$ term in the numerator is a $\Gamma ^{\mathrm{final}}$-covariant called a numerator polynomial of degree $\nu _k$ and is noted $E^{(\nu _k)}\left( \Gamma ^{\mathrm{final}};\Gamma ^{\mathrm{initial}}\right)$ (when $\Gamma ^{\mathrm{final}}$ is degenerate, $E^{(\nu _k)}\left( \Gamma ^{\mathrm{final}};\Gamma ^{\mathrm{initial}}\right)$ will be a vector gathering all the $\Gamma ^{\mathrm{final}},i$-covariant numerator polynomials of degree $\nu _k$ for $i\in \{1,\ldots , \left[ \Gamma ^{\mathrm{final}}\right] \}$). According to the expression, Eq. 39, $D$ denominator polynomials and $N$ numerator polynomials are associated to the generating function, $M^{\mathrm {T}_d}\left( \Gamma ^{\mathrm{final}} ;\Gamma ^{\mathrm{initial}} ; t \right)$.

We will closely follow the article of Patera, Sharp and Winternitz [58] for the notation for denominator and numerator polynomials, using $\alpha , \beta , \gamma$ symbols for a chosen basis of each irrep. However, their table for octahedral tensors contains two errors for the degree eight $E^{(8)}\left( \Gamma _4;\Gamma _4\right)$ and degree seven $E^{(7)}\left( \Gamma _5;\Gamma _4\right)$ numerator polynomials. With the definitions of polynomials given in ref. [58], the following relation hold:

$$\begin{aligned} E^{(8)}\left( \Gamma _4;\Gamma _4\right) _i&= I^{(2)}\left( \Gamma _4\right) E^{(6)}\left( \Gamma _4;\Gamma _4\right) _i\nonumber \\&-\frac{1}{2}I^{(2)}\left( \Gamma _4\right) ^2E^{(4)}\left( \Gamma _4;\Gamma _4\right) _i\nonumber \\&+\frac{1}{2}I^{(4)}\left( \Gamma _4\right) E^{(4)}\left( \Gamma _4;\Gamma _4\right) _i, \end{aligned}$$

(40)

where the index $i$ stands either for $x$, $y$ or $z$. The relation Eq. 40 indicates that the polynomial of degree eight $E^{(8)}\left( \Gamma _4;\Gamma _4\right)$ has a decomposition in terms of polynomials that are elements of the integrity basis associated to $M^{\mathrm {T}_d}\left( \Gamma _4;\Gamma _4;t\right)$. As a consequence, $E^{(8)}\left( \Gamma _4;\Gamma _4\right)$ does not enter the integrity basis.

The same is true for $E^{(7)}\left( \Gamma _5;\Gamma _4\right)$ and the integrity basis associated to $M^{\mathrm {T}_d}\left( \Gamma _5;\right.$ $\left. \Gamma _4;t\right)$ due to following relation:

$$\begin{aligned} E^{(7)}\left( \Gamma _5;\Gamma _4\right) _i&= I^{(2)}\left( \Gamma _4\right) E^{(5)}\left( \Gamma _5;\Gamma _4\right) _i\nonumber \\&-\frac{1}{2}I^{(2)}\left( \Gamma _4\right) ^2E^{(3)}\left( \Gamma _5;\Gamma _4\right) _i\nonumber \\&+\frac{1}{2}I^{(4)}\left( \Gamma _4\right) E^{(3)}\left( \Gamma _5;\Gamma _4\right) _i. \end{aligned}$$

(41)

A complete list of tables of both denominator and numerator polynomials for all the initial $\Gamma ^{\mathrm{initial}}$ and final $\Gamma ^{\mathrm{final}}$ irreps is given in the next sections.

1.1 $\Gamma ^{\mathrm{initial}}=A_1$ irreducible representation

The denominator is $\mathcal {D}\left( A_1;t\right) = 1-t$. The corresponding denominator polynomial of degree one is $I^{\left( 1\right) }\left( A_1\right) = \alpha$. The only non-zero numerator polynomial is $\mathcal {N}\left( A_1;A_1;t\right) =1$.

1.2 $\Gamma ^{\mathrm{initial}}=A_2$ irreducible representation

The denominator is $\mathcal {D}\left( A_2;t\right) =1-t^2$. The corresponding denominator polynomial of degree two is $I^{\left( 2\right) }\left( A_2\right) =\alpha ^2$. Two numerator polynomials are non-zero: $\mathcal {N}\left( A_1;A_2;t\right) =1$ and $\mathcal {N}\left( A_2;A_2;t\right) =t$. The $A_2$-covariant numerator polynomial of degree one is

$$\begin{aligned} E^{\left( 1\right) }\left( A_2;A_2\right) =\alpha . \end{aligned}$$

1.3 $\Gamma ^{\mathrm{initial}}=E$ irreducible representation

The denominator is $\mathcal {D}\left( E;t\right) =\left( 1-t^2\right) \left( 1-t^3\right)$. The denominator polynomial of degree two is $I^{\left( 2\right) }\left( E\right) =\frac{\alpha ^2+\beta ^2}{\sqrt{2}}$ and the denominator polynomial of degree three is $I^{\left( 3\right) }\left( E\right) =\frac{-\alpha ^3+3\alpha \beta ^2}{2}$. Three numerator polynomials are non-zero: $\mathcal {N}\left( A_1;E;t\right) =1$, $\mathcal {N}\left( A_2;E;t\right) =t^3$, and $\mathcal {N}\left( E;E;t\right) =t+t^2$. The $A_2$-covariant numerator polynomial of degree three is

$$\begin{aligned} E^{\left( 3\right) }\left( A_2;E\right) =\frac{-3\alpha ^2\beta +\beta ^3}{2}, \end{aligned}$$

and the two $E$-covariant numerator polynomials of degree one and two are

$$\begin{aligned} E^{\left( 1\right) }\left( E;E\right)&= \left( \begin{array}{c}\alpha \\ \beta \end{array}\right) ,\\ E^{\left( 2\right) }\left( E;E\right)&= \frac{1}{\sqrt{2}}\left( \begin{array}{c}-\alpha ^2+\beta ^2\\ 2\alpha \beta \end{array}\right) . \end{aligned}$$

1.4 $\Gamma ^{\mathrm{initial}}=F_1$ irreducible representation

The denominator is $\mathcal {D}\left( F_1;t\right) =\left( 1-t^2\right) \left( 1-t^4\right) \left( 1-t^6\right)$. The denominator polynomial of degree two is $I^{\left( 2\right) }\left( F_1\right) =\frac{\alpha ^2+\beta ^2+\gamma ^2}{\sqrt{3}}$, the denominator polynomial of degree four is $I^{\left( 4\right) }\left( F_1\right) =\frac{\alpha ^4+\beta ^4+\gamma ^4}{\sqrt{3}}$ and the denominator polynomial of degree six is $I^{\left( 6\right) }\left( F_1\right) =\frac{\alpha ^6+\beta ^6+\gamma ^6}{\sqrt{3}}$. The numerator polynomials are $\mathcal {N}\left( A_1;F_1;t\right) =1+t^9$, $\mathcal {N}\left( A_2;F_1;t\right) =t^3+t^6$, $\mathcal {N}\left( E;F_1;t\right) =t^2+t^4+t^5+t^7$, $\mathcal {N}\left( F_1;F_1;t\right) =t+t^3+t^4+t^5+t^6+t^8$, and $\mathcal {N}\left( F_2;F_1;t\right) =t^2+t^3+t^4+t^5+t^6+t^7$. The invariant numerator polynomial of degree nine is

$$\begin{aligned} E^{\left( 9\right) }\left( A_1;F_1\right) =\frac{1}{\sqrt{6}} \alpha \beta \gamma \left( \alpha ^2-\beta ^2\right) \left( \beta ^2-\gamma ^2\right) \left( \gamma ^2-\alpha ^2\right) , \end{aligned}$$

the two $A_2$-covariant numerator polynomials of degree three and six are

$$\begin{aligned} E^{\left( 3\right) }\left( A_2;F_1\right)&= \alpha \beta \gamma ,\\ E^{\left( 6\right) }\left( A_2;F_1\right)&= \frac{1}{\sqrt{6}}\left( \alpha ^2-\beta ^2\right) \left( \beta ^2-\gamma ^2\right) \left( \gamma ^2-\alpha ^2\right) , \end{aligned}$$

the four $E$-covariant numerator polynomials of degree two, four, five, and seven are:

$$\begin{aligned} E^{\left( 2\right) }\left( E;F_1\right)&= \frac{1}{\sqrt{6}}\left( \begin{array}{c} \alpha ^2+\beta ^2-2\gamma ^2 \\ \sqrt{3}\left( -\alpha ^2+\beta ^2\right) \end{array}\right) ,\\ E^{\left( 4\right) }\left( E;F_1\right)&= \frac{1}{\sqrt{6}}\left( \begin{array}{c} \alpha ^4+\beta ^4-2\gamma ^4 \\ \sqrt{3}\left( -\alpha ^4+\beta ^4\right) \end{array}\right) , \\ E^{\left( 5\right) }\left( E;F_1\right)&= \frac{1}{\sqrt{6}}\alpha \beta \gamma \left( \begin{array}{c} \sqrt{3}\left( \alpha ^2-\beta ^2\right) \\ \alpha ^2+\beta ^2-2\gamma ^2 \end{array}\right) ,\\ E^{\left( 7\right) }\left( E;F_1\right)&= \frac{1}{\sqrt{6}}\alpha \beta \gamma \left( \begin{array}{c} \sqrt{3}\left( \alpha ^4-\beta ^4\right) \\ \alpha ^4+\beta ^4-2\gamma ^4\end{array}\right) , \end{aligned}$$

the six $F_1$-covariant numerator polynomials of degree one, three, four, five, six, and eight are

$$\begin{aligned} E^{\left( 1\right) }\left( F_1;F_1\right)&= \left( \begin{array}{c} \alpha \\ \beta \\ \gamma \end{array}\right) ,\\ E^{\left( 3\right) }\left( F_1;F_1\right)&= \left( \begin{array}{c} \alpha ^3 \\ \beta ^3 \\ \gamma ^3 \end{array}\right) ,\\ E^{\left( 4\right) }\left( F_1;F_1\right)&= \frac{1}{\sqrt{2}}\left( \begin{array}{c} \left( \beta ^2-\gamma ^2\right) \beta \gamma \\ \left( \gamma ^2-\alpha ^2\right) \gamma \alpha \\ \left( \alpha ^2-\beta ^2\right) \alpha \beta \end{array}\right) ,\\ E^{\left( 5\right) }\left( F_1;F_1\right)&= \left( \begin{array}{c} \alpha ^5 \\ \beta ^5\\ \gamma ^5\end{array}\right) ,\\ E^{\left( 6\right) }\left( F_1;F_1\right)&= \frac{1}{\sqrt{2}}\left( \begin{array}{c} \left( \beta ^4-\gamma ^4\right) \beta \gamma \\ \left( \gamma ^4-\alpha ^4\right) \gamma \alpha \\ \left( \alpha ^4-\beta ^4\right) \alpha \beta \end{array}\right) ,\\ E^{\left( 8\right) }\left( F_1;F_1\right)&= \frac{1}{\sqrt{2}} \alpha \beta \gamma \left( \begin{array}{c} \left( \beta ^4-\gamma ^4\right) \alpha \\ \left( \gamma ^4-\alpha ^4\right) \beta \\ \left( \alpha ^4-\beta ^4\right) \gamma \end{array}\right) , \end{aligned}$$

the six $F_2$-covariant numerator polynomials of degree two, three, four, five, six, and seven are

$$\begin{aligned} E^{\left( 2\right) }\left( F_2;F_1\right)&= \left( \begin{array}{c} \beta \gamma \\ \gamma \alpha \\ \alpha \beta \end{array}\right) \\ E^{\left( 3\right) }\left( F_2;F_1\right)&= \frac{1}{\sqrt{2}}\left( \begin{array}{c} \left( \beta ^2-\gamma ^2\right) \alpha \\ \left( \gamma ^2-\alpha ^2\right) \beta \\ \left( \alpha ^2-\beta ^2\right) \gamma \end{array}\right) ,\\ E^{\left( 4\right) }\left( F_2;F_1\right)&= \alpha \beta \gamma \left( \begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right) ,\\ E^{\left( 5\right) }\left( F_2;F_1\right)&= \frac{1}{\sqrt{2}}\left( \begin{array}{c} \left( \beta ^4-\gamma ^4\right) \alpha \\ \left( \gamma ^4-\alpha ^4\right) \beta \\ \left( \alpha ^4-\beta ^4\right) \gamma \end{array}\right) ,\\ E^{\left( 6\right) }\left( F_2;F_1\right)&= \alpha \beta \gamma \left( \begin{array}{c}\alpha ^3 \\ \beta ^3\\ \gamma ^3 \end{array}\right) , \\ E^{\left( 7\right) }\left( F_2;F_1\right)&= \frac{1}{\sqrt{2}} \alpha \beta \gamma \left( \begin{array}{c} \left( \beta ^2-\gamma ^2\right) \beta \gamma \\ \left( \gamma ^2-\alpha ^2\right) \alpha \gamma \\ \left( \alpha ^2-\beta ^2\right) \alpha \beta \end{array}\right) . \end{aligned}$$

1.5 $\Gamma ^{\mathrm{initial}}=F_2$ irreducible representation

The denominator is $\mathcal {D}\left( F_2;t\right) =\left( 1-t^2\right) \left( 1-t^3\right) \left( 1-t^4\right)$. The denominator polynomial of degree two is $I^{\left( 2\right) }\left( F_2\right) =\frac{\alpha ^2+\beta ^2+\gamma ^2}{\sqrt{3}}$, the denominator polynomial of degree three is $I^{\left( 3\right) }\left( F_2\right) =\alpha \beta \gamma$ and the denominator polynomial of degree four is $I^{\left( 4\right) }\left( F_2\right) =\frac{\alpha ^4+\beta ^4+\gamma ^4}{\sqrt{3}}$. The numerator polynomials are $\mathcal {N}\left( A_1;F_2;t\right) =1$, $\mathcal {N}\left( A_2;F_2;t\right) =t^6$, $\mathcal {N}\left( E;F_2;t\right) =t^2+t^4$, $\mathcal {N}\left( F_1;F_2;t\right) =t^3+t^4+t^5$, and $\mathcal {N}\left( F_2;F_2;t\right) =t+t^2+t^3$. The $A_2$-covariant numerator polynomial of degree six is

$$\begin{aligned} E^{\left( 6\right) }\left( A_2;F_2\right) = \frac{1}{\sqrt{6}} \left( \alpha ^2-\beta ^2\right) \left( \beta ^2-\gamma ^2\right) \left( \gamma ^2-\alpha ^2\right) , \end{aligned}$$

the two $E$-covariant numerator polynomials of degree two and four are

$$\begin{aligned} E^{\left( 2\right) }\left( E;F_2\right)&= \frac{1}{\sqrt{6}} \left( \begin{array}{c} \alpha ^2+\beta ^2-2\gamma ^2 \\ \sqrt{3}\left( -\alpha ^2+\beta ^2\right) \end{array}\right) ,\\ E^{\left( 4\right) }\left( E;F_2\right)&= \frac{1}{\sqrt{6}} \left( \begin{array}{c} \alpha ^4+\beta ^4-2\gamma ^4 \\ \sqrt{3}\left( -\alpha ^4+\beta ^4\right) \end{array}\right) , \end{aligned}$$

the three $F_1$-covariant numerator polynomials of degree three, four and five are

$$\begin{aligned} E^{\left( 3\right) }\left( F_1;F_2\right)&= \frac{1}{\sqrt{2}} \left( \begin{array}{c} \left( \beta ^2-\gamma ^2\right) \alpha \\ \left( \gamma ^2-\alpha ^2\right) \beta \\ \left( \alpha ^2-\beta ^2\right) \gamma \end{array}\right) ,\\ E^{\left( 4\right) }\left( F_1;F_2\right)&= \frac{1}{\sqrt{2}} \left( \begin{array}{c} \left( \beta ^2-\gamma ^2\right) \beta \gamma \\ \left( \gamma ^2-\alpha ^2\right) \gamma \alpha \\ \left( \alpha ^2-\beta ^2\right) \alpha \beta \end{array}\right) ,\\ E^{\left( 5\right) }\left( F_1;F_2\right)&= \frac{1}{\sqrt{2}} \left( \begin{array}{c} \left( \beta ^2-\gamma ^2\right) \alpha ^3 \\ \left( \gamma ^2-\alpha ^2\right) \beta ^3\\ \left( \alpha ^2-\beta ^2\right) \gamma ^3 \end{array}\right) , \end{aligned}$$

the three $F_2$-covariant numerator polynomials of degree one, two, and three are

$$\begin{aligned} E^{\left( 1\right) }\left( F_2;F_2\right)&= \left( \begin{array}{c} \alpha \\ \beta \\ \gamma \end{array}\right) ,\\ E^{\left( 2\right) }\left( F_2;F_2\right)&= \left( \begin{array}{c}\beta \gamma \\ \gamma \alpha \\ \alpha \beta \end{array}\right) ,\\ E^{\left( 3\right) }\left( F_2;F_2\right)&= \left( \begin{array}{c} \alpha ^3 \\ \beta ^3 \\ \gamma ^3 \end{array}\right) . \end{aligned}$$

Appendix 2: Application of the integrity basis for $F_2$-covariant polynomials: representation of the electric dipole moment surface of a tetrahedral $\mathrm {XY_4}$ molecule

1.1 Introduction

Appendix 2 gives an application of the integrity basis for $F_2$-covariant polynomials of tetrahedral $\mathrm {XY_4}$ molecules. The integrity basis determined in this paper contains the denominator polynomials $f_i$, $1\le i\le 9$, listed in the main text and the auxiliary numerators published in the file symmetries_A1_A2_E_F1_F2.txt available as supplemental material [54]. This example can be transposed to any other final irrep $\Gamma ^{\mathrm{final}}$.

The electric dipole moment surface of a tetrahedral $\mathrm {XY_4}$ molecule can be built as a linear combination of $F_2$-covariant polynomials of total degree less than $d_\mathrm {max}$ in the coordinates that span the representation, $\Gamma ^{\mathrm{initial}}$, of Eq. 23. The integer $d_\mathrm {max}$ is the order of the expansion. The generating function for the number of $F_2$-covariant polynomials built from this representation reads (see Eqs. 35–37):

$$\begin{aligned} \frac{2t + 5t^2 \!+\! 12t^3 + 23t^4 + 41t^5 + 60t^6+ 71t^7 + 71t^8 + 60t^9 + 45t^{10} + 27t^{11} + 12t^{12} + 3t^{13}}{\left( 1-t\right) \left( 1-t^2\right) ^3\left( 1-t^3\right) ^3\left( 1-t^4\right) ^2}, \end{aligned}$$

whose Taylor expansion up to order four is given by:

$$\begin{aligned} 2t+7t^2+25t^3+69t^4+\cdots . \end{aligned}$$

(42)

The coefficients in Eq. 42 mean that there are $2$ (respectively $7$, $25$, and $69$) linearly independent $F_2,\alpha$-covariant polynomials of degree one (respectively two, three, and four), $\alpha \in \{x,y,z\}$. We now detail the construction of these $103$ $F_2,x$ polynomials. The $F_2,y$ and $F_2,z$ polynomials may be built using the same procedure.

The expansion of the $F_2, x$-EDMS up to order four is a linear combination of $103$ $F_2, x$-polynomials:

$$\begin{aligned}&{\mu _{F_2,x}\left( S_1,S_{2a},S_{2b},S_{3x},S_{3y},S_{3z},S_{4x},S_{4y},S_{4z}\right) } \nonumber \\&\quad =\sum _{i=1}^{103} c_i^{F_2,x} \times p_{i}^{F_2,x}\left( S_1,S_{2a},S_{2b},S_{3x},S_{3y},S_{3z},S_{4x},S_{4y},S_{4z}\right) . \end{aligned}$$

(43)

The coefficients $c_i^{F_2,x}$ of Eq. 43 are to be determined by fitting the expression to either experimental or ab initio data. We know that the $103$, $F_2,x$-polynomials, $p_{i}^{F_2,x}$, can be written as a product of denominator polynomials powered to any positive integer value, and a single numerator polynomial. So, the polynomials that enter the expansion of the $F_2,x$ component of the EDMS can all be taken of the form:

$$\begin{aligned} \varphi _{k,l}^{F_2,x} \times f_1^{n_1}f_2^{n_2}\cdots f_9^{n_9}, \end{aligned}$$

(44)

where the $(\varphi _{k,l}^{F_2,x})_{1\le l \le n_k^{F_2}}$ denotes the numerator polynomials of degree $k$, (we change the notation with respect to the main text to include explicitly the degree $k$). Their numbers, $n_k^{F_2}$, are given in the column labelled $F_2$ of Table 6. Sets of linearly independent $p_{i}^{F_2,x}$ are listed below by degrees. We recall that $f_1$ is a polynomial of degree one, $f_2$, $f_3$, and $f_4$ are three polynomials of degree two, $f_5$, $f_6$, and $f_7$ are three polynomials of degree three, and $f_8$, $f_9$ are two polynomials of degree four.

1.2 Degree one

The $2$ $F_2,x$ linearly independent polynomials of total degree one compatible with Eq. 44 are $p_{1}^{F_2,x}=\varphi _{1,1}^{F_2,x}$ and $p_{2}^{F_2,x}=\varphi _{1,2}^{F_2,x}$.

1.3 Degree two

The $7$ $F_2,x$ linearly independent polynomials of total degree two compatible with Eq. 44 are:

$$\begin{aligned} \begin{array}{rclrclrclrclrcl} p_{3}^{F_2,x}&{}=&{}\varphi _{2,1}^{F_2,x}, &{} p_{4}^{F_2,x}&{}=&{}\varphi _{2,2}^{F_2,x}, &{} p_{5}^{F_2,x}&{}=&{}\varphi _{2,3}^{F_2,x}, &{} p_{6}^{F_2,x}&{}=&{}\varphi _{2,4}^{F_2,x}, &{} p_{7}^{F_2,x}&{}=&{}\varphi _{2,5}^{F_2,x}, \\ p_{8}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x}f_1, &{} p_{9}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x}f_1. \end{array} \end{aligned}$$

1.4 Degree three

The $25$ $F_2,x$ linearly independent polynomials of total degree three compatible with Eq. 44 are:

$$\begin{aligned} \begin{array}{rclrclrclrclrcl} p_{10}^{F_2,x}&{}=&{}\varphi _{3,1}^{F_2,x}, &{} p_{11}^{F_2,x}&{}=&{}\varphi _{3,2}^{F_2,x}, &{} p_{12}^{F_2,x}&{}=&{}\varphi _{3,3}^{F_2,x}, &{} p_{13}^{F_2,x}&{}=&{}\varphi _{3,4}^{F_2,x}, &{} p_{14}^{F_2,x}&{}=&{}\varphi _{3,5}^{F_2,x}, \\ p_{15}^{F_2,x}&{}=&{}\varphi _{3,6}^{F_2,x}, &{} p_{16}^{F_2,x}&{}=&{}\varphi _{3,7}^{F_2,x}, &{} p_{17}^{F_2,x}&{}=&{}\varphi _{3,8}^{F_2,x}, &{} p_{18}^{F_2,x}&{}=&{}\varphi _{3,9}^{F_2,x}, &{} p_{19}^{F_2,x}&{}=&{}\varphi _{3,{10}}^{F_2,x}, \\ p_{20}^{F_2,x}&{}=&{}\varphi _{3,{11}}^{F_2,x}, &{} p_{21}^{F_2,x}&{}=&{}\varphi _{3,{12}}^{F_2,x}, &{} p_{22}^{F_2,x}&{}=&{}\varphi _{2,1}^{F_2,x} f_1, &{} p_{23}^{F_2,x}&{}=&{}\varphi _{2,2}^{F_2,x} f_1, &{} p_{24}^{F_2,x}&{}=&{}\varphi _{2,3}^{F_2,x} f_1, \\ p_{25}^{F_2,x}&{}=&{}\varphi _{2,4}^{F_2,x} f_1, &{} p_{26}^{F_2,x}&{}=&{}\varphi _{2,5}^{F_2,x} f_1, &{} p_{27}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_1^2, &{} p_{28}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_1^2, &{} p_{29}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_2, \\ p_{30}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_2, &{} p_{31}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_3, &{} p_{32}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_3, &{} p_{33}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_4, &{} p_{34}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_4. \end{array} \end{aligned}$$

1.5 Degree four

The $69$ $F_2,x$ linearly independent polynomials of total degree four compatible with Eq. 44 are:

$$\begin{aligned} \begin{array}{rclrclrclrclrcl} p_{35}^{F_2,x}&{}=&{}\varphi _{4,1}^{F_2,x}, &{} p_{36}^{F_2,x}&{}=&{}\varphi _{4,2}^{F_2,x}, &{} p_{37}^{F_2,x}&{}=&{}\varphi _{4,3}^{F_2,x}, &{} p_{38}^{F_2,x}&{}=&{}\varphi _{4,4}^{F_2,x}, &{} p_{39}^{F_2,x}&{}=&{}\varphi _{4,5}^{F_2,x}, \\ p_{40}^{F_2,x}&{}=&{}\varphi _{4,6}^{F_2,x}, &{} p_{41}^{F_2,x}&{}=&{}\varphi _{4,7}^{F_2,x}, &{} p_{42}^{F_2,x}&{}=&{}\varphi _{4,8}^{F_2,x}, &{} p_{43}^{F_2,x}&{}=&{}\varphi _{4,9}^{F_2,x}, &{} p_{44}^{F_2,x}&{}=&{}\varphi _{4,10}^{F_2,x}, \\ p_{45}^{F_2,x}&{}=&{}\varphi _{4,11}^{F_2,x}, &{} p_{46}^{F_2,x}&{}=&{}\varphi _{4,12}^{F_2,x}, &{} p_{47}^{F_2,x}&{}=&{}\varphi _{4,13}^{F_2,x}, &{} p_{48}^{F_2,x}&{}=&{}\varphi _{4,14}^{F_2,x}, &{} p_{49}^{F_2,x}&{}=&{}\varphi _{4,15}^{F_2,x}, \\ p_{50}^{F_2,x}&{}=&{}\varphi _{4,16}^{F_2,x}, &{} p_{51}^{F_2,x}&{}=&{}\varphi _{4,17}^{F_2,x}, &{} p_{52}^{F_2,x}&{}=&{}\varphi _{4,18}^{F_2,x}, &{} p_{53}^{F_2,x}&{}=&{}\varphi _{4,19}^{F_2,x}, &{} p_{54}^{F_2,x}&{}=&{}\varphi _{4,20}^{F_2,x}, \\ p_{55}^{F_2,x}&{}=&{}\varphi _{4,21}^{F_2,x}, &{} p_{56}^{F_2,x}&{}=&{}\varphi _{4,22}^{F_2,x}, &{} p_{57}^{F_2,x}&{}=&{}\varphi _{4,23}^{F_2,x}, &{} p_{58}^{F_2,x}&{}=&{}\varphi _{2,1}^{F_2,x} f_2, &{} p_{59}^{F_2,x}&{}=&{}\varphi _{2,2}^{F_2,x} f_2, \\ p_{60}^{F_2,x}&{}=&{}\varphi _{2,3}^{F_2,x} f_2, &{} p_{61}^{F_2,x}&{}=&{}\varphi _{2,4}^{F_2,x} f_2, &{} p_{62}^{F_2,x}&{}=&{}\varphi _{2,5}^{F_2,x} f_2, &{} p_{63}^{F_2,x}&{}=&{}\varphi _{2,1}^{F_2,x} f_3, &{} p_{64}^{F_2,x}&{}=&{}\varphi _{2,2}^{F_2,x} f_3, \\ p_{65}^{F_2,x}&{}=&{}\varphi _{2,3}^{F_2,x} f_3, &{} p_{66}^{F_2,x}&{}=&{}\varphi _{2,4}^{F_2,x} f_3, &{} p_{67}^{F_2,x}&{}=&{}\varphi _{2,5}^{F_2,x} f_3, &{} p_{68}^{F_2,x}&{}=&{}\varphi _{2,1}^{F_2,x} f_5, &{} p_{69}^{F_2,x}&{}=&{}\varphi _{2,2}^{F_2,x} f_5, \\ p_{70}^{F_2,x}&{}=&{}\varphi _{2,3}^{F_2,x} f_5, &{} p_{71}^{F_2,x}&{}=&{}\varphi _{2,4}^{F_2,x} f_5, &{} p_{72}^{F_2,x}&{}=&{}\varphi _{2,5}^{F_2,x} f_5, &{} p_{73}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_5, &{} p_{74}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_5, \\ p_{75}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_6, &{} p_{76}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_6, &{} p_{77}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_7, &{} p_{78}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_7, &{} p_{79}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_1 f_2, \\ p_{80}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_1 f_2, &{} p_{81}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_1 f_3, &{} p_{82}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_1 f_3, &{} p_{83}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_1 f_4, &{} p_{84}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_1 f_4, \\ p_{85}^{F_2,x}&{}=&{}\varphi _{3,1}^{F_2,x} f_1, &{} p_{86}^{F_2,x}&{}=&{}\varphi _{3,2}^{F_2,x} f_1, &{} p_{87}^{F_2,x}&{}=&{}\varphi _{3,3}^{F_2,x} f_1, &{} p_{88}^{F_2,x}&{}=&{}\varphi _{3,4}^{F_2,x} f_1, &{} p_{89}^{F_2,x}&{}=&{}\varphi _{3,5}^{F_2,x} f_1, \\ p_{90}^{F_2,x}&{}=&{}\varphi _{3,6}^{F_2,x} f_1, &{} p_{91}^{F_2,x}&{}=&{}\varphi _{3,7}^{F_2,x} f_1, &{} p_{92}^{F_2,x}&{}=&{}\varphi _{3,8}^{F_2,x} f_1, &{} p_{93}^{F_2,x}&{}=&{}\varphi _{3,9}^{F_2,x} f_1, &{} p_{94}^{F_2,x}&{}=&{}\varphi _{3,10}^{F_2,x} f_1, \\ p_{95}^{F_2,x}&{}=&{}\varphi _{3,11}^{F_2,x} f_1, &{} p_{96}^{F_2,x}&{}=&{}\varphi _{3,12}^{F_2,x} f_1, &{} p_{97}^{F_2,x}&{}=&{}\varphi _{2,1}^{F_2,x} f_1^2, &{} p_{98}^{F_2,x}&{}=&{}\varphi _{2,2}^{F_2,x} f_1^2, &{} p_{99}^{F_2,x}&{}=&{}\varphi _{2,3}^{F_2,x} f_1^2, \\ p_{100}^{F_2,x}&{}=&{}\varphi _{2,4}^{F_2,x} f_1^2, &{} p_{101}^{F_2,x}&{}=&{}\varphi _{2,5}^{F_2,x} f_1^2, &{} p_{102}^{F_2,x}&{}=&{}\varphi _{1,1}^{F_2,x} f_1^3, &{} p_{103}^{F_2,x}&{}=&{}\varphi _{1,2}^{F_2,x} f_1^3. \end{array} \end{aligned}$$