On mu-Symmetric Polynomials

Suppose $`P(x)\in \ZZ[x]`$ is a polynomial with $`m`$ distinct complex roots $`r_1\dd r_m`$ where $`r_i`$ has multiplicity $`\mu_i`$. Write $`\bfmu=(\mu_1\dd \mu_m)`$ where we may assume $`\mu_1\ge \mu_2\ge\cdots\ge\mu_m\ge1`$. Thus $`n=\sum_{i=1}^m \mu_i`$ is the degree of $`P(x)`$. Consider the following function of the roots

\dplus(P(x)) \as \prod_{1\le i<j\le m}
            (r_i-r_j)^{\mu_i+\mu_j}.

Call this the root function. The form of this root function¹ was introduced by Becker et al in their complexity analysis of a root clustering algorithm. The origin of this paper was to try to prove that $`\dplus(P(x))`$ is a rational function in the coefficients of $`P(x)`$. This result is needed for obtaining an explicit upper bound on the complexity of the algorithm on integer polynomials . This application is detailed in our companion paper .

We may write “$`\dplus(\bfmu)`$” instead of $`\dplus(P(x))`$ since the expression in terms of the roots $`\bfr=(r_1\dd r_m)`$ depends only on the multiplicity structure $`\bfmu`$. For example, if $`\bfmu=(2,1)`$ then $`\dplus(\bfmu)=(r_1-r_2)^3`$ and this turns out to be

\left[a_1^3-(9/2)a_0a_1a_2+(27/2)a_0^2a_3\right]/a_0^3

when $`P(x)=\sum_{i=0}^3 a_{3-i}x^i`$. More generally, for any function $`F(\bfr)=F(r_1\dd r_m)`$, we ask whether evaluating $`F`$ at the $`m`$ distinct roots of a polynomial $`P(x)`$ with multiplicity structure $`\bfmu`$ is rational in the coefficients of $`P(x)`$.

The Fundamental Theorem of Symmetric Functions gives a partial answer: if $`F(\bfr)`$ is a symmetric polynomial then $`F(\bfr)`$ is a rational function in the coefficients of $`P(x)`$. This result does not exploit knowledge of the multiplicity structure $`\bfmu`$ of $`P(x)`$. We want a natural definition of “$`\bfmu`$-symmetry” such that the following property is true: if $`F(\bfr)`$ is $`\bfmu`$-symmetric, then $`F(\bfr)`$ is a rational function in the coefficients of $`P(x)`$. When $`\bfmu=(1\dd 1)`$, i.e., all the roots of $`P(x)`$ are simple, then a $`\bfmu`$-symmetric polynomial is just a symmetric polynomial in the usual sense. So our original goal amounts to proving that $`\dplus(\bfmu)`$ is $`\bfmu`$-symmetric. It is non-trivial to check if any given root function $`F`$ (in particular $`F=\dplus(\bfmu)`$) is $`\bfmu`$-symmetric. We will designed three algorithms for this task. Although we feel that $`\bfmu`$-symmetry is a natural concept, to our knowledge, this has not been systematically studied before.

The rest of this paper is organized as follows. In Section 2, we defined $`\bfmu`$-symmetric polynomials in terms of elementary symmetric polynomials and show some preliminary properties of such polynomials. In Section 3, we proved the $`\bfmu`$-symmetry of $`\dplus`$ for some special $`\bfmu`$. To investigate the $`\bfmu`$-symmetry of $`\dplus`$ in the general case, three algorithms for checking $`\bfmu`$-symmetry are given in Sections 4-6. In Section 8, we show experimental results from our Maple implementation of the three algorithms. We conclude in Section 9.

Throughout the paper, assume $`K`$ is a field of characteristic $`0`$. For our purposes, $`K=\QQ`$ will do. We also fix three sequences of variables

\bfx=(x_1\dd x_n),\quad
        \bfz=(z_1\dd z_n),\quad \bfr=(r_1\dd r_m)

where $`n\ge m\ge 1`$. Intuitively, the $`x_i`$’s are roots (not necessarily distinct), $`z_i`$’s are variables representing the elementary symmetric functions of the roots, and $`r_i`$’s are the distinct roots.

Let $`\bfmu`$ be a partition of $`n`$ with $`m`$ parts. In other words, $`\bfmu=(\mu_1\dd \mu_m)`$ where $`n=\mu_1+\cdots+\mu_m`$ and $`\mu_1\ge \mu_2\ge\cdots\ge\mu_m\ge 1`$. We denote this relation by

\bfmu \vdash n.

We call $`\bfmu`$ an if it has $`m`$ parts. A $`\sigma`$ is any function of the form $`\sigma:\set{x_1\dd x_n}\to\set{r_1\dd r_m}`$. We say $`\sigma`$ is of if $`\#\sigma\inv(r_i)=\mu_i`$ for $`i=1\dd m`$. Throughout the paper, we use $`\#`$ to denote the number of elements in a set, and $`|\cdot|`$ to denote the length of a sequence. In particular, $`|\bfmu|=|\bfr|=m`$. We say $`\sigma`$ is if $`\sigma(x_i)=r_j`$ and $`\sigma(x_{i+1})=r_k`$ implies $`j\le k`$. Clearly the canonical specialization of type $`\bfmu`$ is unique, and we may denote it by $`\sigma_\bfmu`$.

Consider the polynomial rings $`K[\bfx]`$ and $`K[\bfr]`$. Any specialization $`\sigma:\set{x_1\dd x_r}`$ $`\to\set{r_1\dd r_m}`$ can be extended naturally into a $`K`$-homomorphism

\sigma: K[\bfx]\to K[\bfr]

where $`P=P(\bfx)\in K[\bfx]`$ is mapped to $`\sigma(P)= P(\sigma(x_1)\dd \sigma(x_n))`$. When $`\sigma`$ is understood, we may write “$`\ol{P}`$” for the homomorphic image $`\sigma(P)`$.

We denote the ($`i=1\dd n`$) in $`K[\bfx]`$ by $`\e_i=\e_i(\bfx)`$. For instance, _1 && _i=1^n x_i,
_2 && _1i<jn x_ix_j,
& ⋮&
_n && _i=1^n x_i. Also define $`\e_0\as 1`$. Typically, we write $`\ole_i`$ for the $`\sigma_\bfmu`$ specialization of $`e_i`$ when $`\bfmu`$ is understood from the context; thus $`\ole_i=\sigma_{\bfmu}(\e_i)\in K[\bfr]`$. For instance, if $`\bfmu=(2,1)`$ then $`\ole_1=2r_1+r_2`$ and $`\ole_2=r_1^2+2r_1r_2`$.

The key definition is the following: a polynomial $`F\in K[\bfr]`$ is said to be if there is a symmetric polynomial $`\whF\in K[\bfx]`$ such that $`\sigma_\bfmu(\whF)=F`$. We call $`\whF`$ the (or simply “lift”) of $`F`$. If $`\ooF\in K[\bfz]`$ satisfies $`\ooF(\e_1\dd \e_n)=\whF(\bfx)`$ then we call $`\ooF`$ the of $`F`$.

We have this consequence of the Fundamental Theorem on Symmetric Functions:

Assume

P(x)=\sum_{i=0}^nc_ix^{n-i}\in K[x]

has $`m`$ distinct roots $`\bfrho=(\rho_1\dd \rho_m)`$ of multiplicity $`\bfmu=(\mu_1\dd \mu_m)`$.

If $`F\in K[\bfr]`$ is $`\bfmu`$-symmetric, then $`F(\bfrho)`$ is an element in $`K`$.

If $`\ooF\in K[\bfz]`$ is the $`\bfmu`$-gist of $`F`$, then

F(\rho_1\dd \rho_m)
                = \ooF\left(-c_1/c_0\dd(-1)^nc_n/c_0\right).

F() &=& _(()) & (by definition of $`\bfmu`$-symmetry)
&=& _((e_1e_n)) & (by the Fundamental Theorem of Symmetric 
&&&  Functions, as $`\whF`$ is symmetric)
&=& (_1_n) & (since $`\ole_i =\sigma_\bfmu(e_i)`$)
F() &=& (_1()_n())
&=& (-c_1/c_0 (-1)^n c_n/c_0) & (by Vieta’s formula for roots) This proves the formula in (ii). The assertion of (i) follows from the fact that $`\ooF\in K[\bfz]`$ and $`c_i`$’s belong to $`K`$.

\ooF(-c_1/c_0,c_2/c_0, -c_3/c_0)=
            \left(-c_1/c_0\right)^2-c_2/c_0\in K

We want to study the lift $`\whF\in K[\bfx]`$ of a $`\bfmu`$-symmetric polynomial $`F\in K[\bfr]`$ of total degree $`\delta`$. If we write $`F`$ as the sum of its homogeneous parts, $`F=F_1+\cdots+ F_\delta`$, then $`\whF=\whF_1+\cdots+\whF_\delta`$. Hence, we may restrict $`F`$ to be homogeneous.

Next consider a polynomial $`H(\bfz)\in K[\bfz]`$. Suppose there is a

\omega: \set{z_1\dd z_n} \to \NN=\set{1,2,\ldots}

then for any term $`t=\prod_{i=1}^n z_i^{d_i}`$, its is $`\sum_{i=1}^n d_i \omega(z_i)`$. Normally, $`\omega(z_i)=1`$ for all $`i`$; but in this paper, we are also interested in the weight function where $`\omega(z_i)=i`$. For short, we simply call this $`\omega`$-degree of $`t`$ its , . The weighted degree of a polynomial $`H(\bfz)`$ is just the maximum weighted degree of terms in its support, . A polynomial $`H(\bfz)`$ is said to be if all of its terms have the same weighted degree. Note that the weighted degree of a polynomial $`H\in K[\bfz]`$ is the same as the degree of $`H(\e_1\dd\e_n)\in K[\bfx]`$.

The gist $`\ooF`$ of $`F`$ is not unique: for any gist $`\ooF`$, we can decompose it as $`\ooF=\ooF_0+\ooF_1`$ where $`\ooF_0`$ is the weighted homogeneous part of $`\ooF`$ of degree $`\delta`$, and $`\ooF_1\as \ooF-\ooF_0`$. Then $`\ooF(\ole_1\dd\ole_n)=F`$ implies that $`\ooF_0(\ole_1\dd\ole_n)=F`$ and $`\ooF_1(\ole_1\dd\ole_n)=0`$. We can always omit $`\ooF_1`$ from the gist of $`F`$. We shall call any polynomial $`H(\bfz)\in K[\bfz]`$ a if $`H(\ole_1\dd\ole_n)=0`$. Thus, $`\ooF_1`$ is a $`\bfmu`$-constraint.

It follows that when trying to check if $`F`$ is $`\bfmu`$-symmetric, it is sufficient to look for gists $`\ooF`$ among weighted homogeneous polynomials of the same degree as $`F`$, i.e., $`\delta`$. But even this restriction does not guarantee uniqueness of the gist of $`F`$ because there could be $`\bfmu`$-constraints of weighted homogeneous degree $`\deg(F)`$. To illustrate this phenomenon, we consider the following example.

H=\ooF-\ooF'=\efrac{8}\Big(z_1^3+8z_3-4z_1z_2)

H(\ole_1\dd\ole_4)
            = \efrac{8}(2r_1+2r_2)^3
        +(2r_1^2r_2+2r_1r_2^2)
        -\efrac{2}(2r_1+2r_2)(r_1^2+4r_1r_2+r_2^2)
        =0.

It is easy to check that the set of all $`\bfmu`$-constraints forms an ideal in $`K[\bfz]`$ which we may call the $`\bfmu`$-ideal, denoted by $`\calJ_{\bfmu}`$.

\begin{align*}
        \left<
        z_1-\ole_1,
        z_2-\ole_2,
        z_3-\ole_3,
        z_4-\ole_4
        \right>
        &=\\
        \left<z_1-(2r_1+2r_2),z_2-(r_1^2\right.&
        \left.+4r_1r_2+r_2^2),
        z_3-(2r_1^2r_2+2r_1r_2^2),
        z_4-r_1^2r_2^2
        \right>.
\end{align*}

Although $`\bfmu`$-symmetric polynomials originated from symmetric polynomials, they differ in many ways as seen in these examples.

A $`\bfmu`$-symmetric polynomial need not be symmetric. Let $`\bfmu=(2,1)`$ and $`n=2+1=3`$. Then $`2r_1+r_2`$ is $`\bfmu`$-symmetric whose lift is $`e_1`$, but it is not symmetric.

A symmetric polynomial need not be $`\bfmu`$-symmetric. Consider the symmetric polynomial $`F=r_1+r_2\in K[r_1,r_2]`$. It is not $`\bfmu`$-symmetric with $`\bfmu=(2,1)`$. If it were, then there is a linear symmetric polynomial $`\whF=ce_1`$ such that $`\sigma_{\bfmu}(\whF)=r_1+r_2`$. But clearly such $`\whF`$ does not exist.

Symmetric polynomials can be $`\bfmu`$-symmetric. Note that $`(r_1-r_2)^2`$ is obviously symmetric in $`K[r_1,r_2]`$. According to , it is also $`\bfmu`$-symmetric for any $`\bfmu=(\mu_1,\mu_2)`$.

In the following we will use this notation: $`[n] \as \set{1\dd n}`$, and let $`[n] \choose k`$ denote the set of all $`k`$-subsets of $`[n]`$. For $`k=0\dd n-2`$, we may define the function S^n_k=S^n_k() _I _ijI (x_i-x_j)^2 called the in $`n`$ variables. By extension, we could also define $`S^n_{n-1}=1`$. When $`k=0`$, we have $`S^n_0= \prod_{i\neq j\in [n]} \Big(x_i-x_j\Big)^2`$. In applications, the $`x_i`$’s are roots of a polynomial $`P(x)`$ of degree $`n`$, and $`S^n_0`$ is the standard discriminant of $`P(x)`$. Clearly $`S^n_{k}`$ is a symmetric polynomial in $`\bfx`$.

Define $`\Delta \as \prod_{1\le i (a) $`\Delta`$ is $`\bfmu`$-symmetric with lift given by

\wh{\Delta} = \efrac{\prod_{i=1}^m\mu_i}
                \cdot S^n_{n-m}

where $`S^n_{n-m}\in K[\bfx]`$ is the $`(n-m)`$-th subdiscriminant.
(b) In particular, when $`m=2`$, we have an explicit formula for the lift of $`\Delta`$:

\wh{\Delta}=\frac{(n-1)e_1^2-2ne_2}{\mu_1\mu_2},

where $`n=\mu_1+\mu_2`$. Let $`\bfmu=(\mu_1\dd \mu_m)`$. Consider the $`m`$-th subdiscriminant $`S^n_m`$ in $`n`$ variables. We may verify that

\sigma_{\bfmu}(S^n_{n-m})=\Delta \cdot \prod_{i=1}^m\mu_i.

This is equivalent to

\sigma_{\bfmu}\left(\frac{1}{\prod_{i=1}^m\mu_i}\cdot
        S^n_{n-m}\right) =\Delta.

Therefore, $`\efrac{\prod_{i=1}^m\mu_i}S^n_{n-m}`$ is the $`\bfmu`$-lift of $`\Delta`$.

To obtain the explicit formula in the case $`m=2`$, consider the symmetric polynomial
$`Q\as\sum_{i

\sigma_{\mu}(Q)=\mu_1\mu_2(r_1-r_2)^2.

Thus, we may choose $`\wh{\Delta}=\frac{(n-1)e_1^2 - 2n e_2}{\mu_1\mu_2}`$.

The following two theorems show the $`\bfmu`$-symmetry of some special $`\dplus`$ polynomials. In other words, they confirmed our conjecture about $`\dplus`$.

There exists $`\ooF_n\in K[\bfz]`$ such that for all $`\bfmu`$ satisfying $`\bfmu=(\mu_1,\mu_2)`$ and $`\mu_1+\mu_2=n`$, we have

\ooF_n(\ole_1,\ole_2)=\dplus(\bfmu).

More explicitly,

$`n`$ is even: $`\ooF_n = \Big( \frac{(n-1)z_1^2-2n z_2}{ \mu_1\mu_2}\Big)^{n/2}`$

$`n`$ is odd: _n &=& ( )^ ( k_1 z_1^3 + k_2 z_1 z_2+k_3 z_3) where $`k_1= \frac{-(n-1)(n-2)}{d}, k_2= \frac{3n(n-2)}{d}, k_3= \frac{-3n^2}{d}`$ and $`d=\mu_1\mu_2(\mu_1-\mu_2)`$. From (b), we know that $`(r_1-r_2)^2`$ is $`\bfmu`$-symmetric for arbitrary $`n`$ and

(r_1-r_2)^2=\frac{(n-1)\ole_1^2-2n\ole_2}{\mu_1\mu_2}.

When $`n`$ is even,

\begin{align*}
    \dplus(\bfmu)=\left((r_1-r_2)^2\right)^{\frac{n}{2}}
        &=\left(\frac{(n-1)\ole_1^2-2n\ole_2}
            {\mu_1\mu_2}\right)^{\frac{n}{2}}\\
        &=\left(\frac{(n-1)\ole_1^2-2n\ole_2}
        {\mu_1\mu_2}\right)^{\frac{n}{2}}=\ooF_n(\ole_1,\ole_2).
\end{align*}

Thus the case for even $`n`$ is proved. It remains to prove the case for odd $`n`$. First, it may be verified that

(r_1-r_2)^3 =k_1\ole_1^3+k_2\ole_1\ole_2+k_3\ole_3,

where

k_1= \frac{-(n-1)(n-2)}{d},\quad
        k_2= \frac{3n(n-2)}{d},\quad
        k_3= \frac{-3n^2}{d}~~\mbox{and}~~
        d=\mu_1\mu_2(\mu_1-\mu_2).

It follows that

\begin{align*}
    \dplus(\bfmu)&=\left((r_1-r_2)^2\right)^{\frac{n-3}{2}}(r_1-r_2)^3\\
        &=\left(\frac{(n-1)\ole_1^2-2n\ole_2}
            {\mu_1\mu_2}\right)^{\frac{n-3}{2}}
          \left(k_1\ole_1^3 + k_2\ole_1\ole_2+k_3\ole_3\right)\\
        &=\left(\frac{(n-1)\ole_1^2-2n\ole_2}
            {\mu_1\mu_2}^{\frac{n}{2}}\right)
        \left(k_1\ole_1^3 +
        k_2\ole_1\ole_2+k_3\ole_3\right)\\
    &=\ooF_n(\ole_1,\ole_2,\ole_3)
\end{align*}

where

k_1= \frac{-(n-1)(n-2)}{d},\quad
        k_2= \frac{3n(n-2)}{d},\quad
        k_3= \frac{-3n^2}{d}~~\mbox{and}~~
        d=\mu_1\mu_2(\mu_1-\mu_2).

Another special case of $`\dplus(\bfmu)`$ is where $`\bfmu=(\mu,\mu\dd \mu)`$.

If all $`\mu_i`$’s are equal to $`\mu`$, then $`\dplus(\bfmu)`$ is $`\bfmu`$-symmetric with lift given by $`\whF_n(\bfx) =\left(\frac{1}{\mu^{m}}\cdot S^n_{n-m}\right)^\mu`$ where $`S^n_{n-m}`$ is given by (a).

Since $`\mu_i=\mu~(1\leq i\leq m)`$,

\dplus(\bfmu)
            =\prod_{i<j}(r_i-r_j)^{2\mu}
        =\left(\prod_{i<j}(r_i-r_j)^2\right)^\mu.

This expression for $`\dplus`$ is $`\bfmu`$-symmetric since $`\prod_{i

The following example shows two ways to compute $`\dplus`$. One is using the definition and the other is using the formula of $`\dplus`$ in coefficients.

\begin{align*}
        \mathring{D}^+(z_1,z_2,z_3,z_4,z_5)=
            &\frac{10125}{4}z_5^2-\frac{11}{2}z_1^2z_2z_3^2
                -3z_1^4z_2z_4+67z_1^3z_3z_4-207z_1^3z_2z_5\\
             &+\frac{2517}{4}z_1z_2^2z_5+171z_1^2z_3z_5
             -\frac{5955}{4}z_2z_3z_5 +\frac{615}{2}z_1z_4z_5\\
             &-184z_2z_4^2+12z_1^5z_5+z_1^4z_3^2+6z_2^2z_3^2
              +\frac{9}{2}z_1z_3^3+48z_2^3z_4\\
             &+\frac{1737}{4}z_3^2z_4+\frac{277}{4}z_1^2z_4^2
             -\frac{1255}{4}z_1z_2z_3z_4.
\end{align*}

In this section, we consider a  basis algorithm to compute the $`\bfmu`$-gist of a given polynomial $`F\in K[\bfr]`$, or detect that it is not $`\bfmu`$-symmetric. In fact, we first generalize our concept of gist: fix an arbitrary (ordered) set

\calD=(d_1\dd d_\ell),\quad d_i\in K[\bfr].

Call $`\calD`$ the basis. If $`F\in K[\bfr]`$ and $`\ooF\in K[\bfy]`$ where $`\bfy=(y_1\dd y_\ell)`$ are $`\ell`$ new variables, then $`\ooF(\bfy)`$ is called a of $`F`$ if $`F(\bfr)=\ooF(d_1\dd d_\ell)`$. Note that if $`\calD=(\ole_1\dd \ole_n)`$ (so $`\ell=n`$) then a $`\calD`$-gist is just a $`\bfmu`$-gist (after renaming $`\bfy`$ to $`\bfz`$).

We now give a method to compute a $`\calD`$-gist of $`F`$ using Gröbner bases. To this end, define the ideal

\calI_\calD \as \bang{v_1\dd v_\ell} \ib K[\bfr,\bfy]

where $`v_i \as y_i-d_i`$. Moreover, let $`\calG_\calD`$ be the  basis of $`\calI_\calD`$ relative to the the term ordering $`\prec_{ry}`$. The ordering is defined as follows:

\bfr^\bfalpha\bfy^\bfbeta \prec_{ry}
                \bfr^{\bfalpha'}\bfy^{\bfbeta'}

iff $`\bfr^\bfalpha \prec_r \bfr^{\bfalpha'}`$ or else $`\bfalpha=\bfalpha'`$ and $`\bfy^\bfbeta \prec_y \bfy^{\bfbeta'}`$. Here $`\prec_r`$ and $`\prec_y`$ are term orderings in $`K[\bfr]`$ and $`K[\bfy]`$ respectively. Note that $`\prec_{ry}`$ is called the lexicographic product of $`\prec_r`$ and $`\prec_y`$ in . We have two useful lemmas. The first is about the ideal $`\calI_\calD`$, and the second about its Gröbner basis $`\calG_\calD`$. For all $`R\in K[\bfy]`$,

R(\bfy)-R(\calD)\in \calI_\calD.

Consider any term $`\bfy^\bfalpha`$ where $`\bfalpha=(\alpha_1\dd\alpha_\ell)`$. Its image in the quotient ring $`K[\bfy]/\calI_\calD`$ is: ^+_ &=& (_i=1^y_i^_i) +_
&=& (_i=1^(d_i+(y_i-d_i))^_i) +_
&=& (_i=1^d_i^_i+_) +_
&=& (_i=1^d_i^_i) +_
&=& ^+_. Thus $`\bfy^\bfalpha-\calD^\bfalpha\in \calI_\calD`$. Since $`R(\bfy)-R(\calD)`$ is a linear combination of $`\bfy^\bfalpha-\calD^\bfalpha`$’s, our lemma is proved.

By a we mean one that is generated by weighted homogeneous polynomials. The following is a generalization of , where the result is stated for homogeneous ideals.

The following is a consequence of : $`\calG_\calD\cap K[\bfy]`$ is a Gröbner basis for the elimination ideal $`\calI_\calD\cap K[\bfy]`$ with respect to the term ordering $`\prec_y`$.

The following is a generalization of Proposition 4 in Cox (except for claims about uniqueness):

Fix the above  basis $`\calG_\calD`$. Let $`R\in K[\bfr,\bfy]`$ be the normal form of $`F\in K[\bfr]`$ relative to $`\calG_\calD`$.

If $`R\in K[\bfy]`$, then $`R`$ is a $`\calD`$-gist of $`F`$.

If $`F`$ has a $`\calD`$-gist, then $`R\in K[\bfy]`$. In the following, we use the specialization $`\sigma: y_i\mapsto d_i`$ for all $`i`$. This induces the homomorphism $`\sigma:K[\bfr,\bfy]\to K[\bfr]`$ taking every polynomial $`f(\bfr,\bfy)`$ in the ideal $`\calI_\calD`$ to $`0`$, i.e., $`\sigma(f)=0`$.

Since $`R`$ is the normal form of $`F`$, $`F-R\in \calI_\calD`$. Thus $`\sigma(F-R)=0`$ or $`\sigma(F)=\sigma(R)`$. But $`F\in K[\bfr]`$ implies $`\sigma(F)=F`$. The assumption that $`R\in K[\bfy]`$ implies that $`\sigma(R) =R(\calD)=R(d_1\dd d_\ell)`$. We conclude that $`R`$ is a $`\calD`$-gist of $`F`$:

F(\bfr)  = R(\calD)

By assumption, $`F`$ has a $`\calD`$-gist $`\ooF\in K[\bfy]`$, i.e., $`\ooF(\calD)=F`$. Let $`\wtR`$ be the normal form of $`\ooF`$. CLAIM: $`R-\wtR\in \calI_\calD`$. To see this, we write $`R-\wtR`$ as a sum

R-\wtR = (R-F)+(F-\ooF) + (\ooF-\wtR).

We only need to verify that each of the three summands belong to $`\calI_\calD`$: in part (i), we noted that $`R-F\in \calI_\calD`$; the third summand $`\ooF-\wtR\in \calI_\calD`$ for the same reason. The second summand $`F-\ooF\in\calI_\calD`$ by an application of . To conclude that $`R\in K[\bfy]`$, we assume (by way of contradiction) that $`R\notin K[\bfy]`$. By our choice of term ordering for $`\calG_\calD`$, we know that $`\lt(R-\wtR)=\lt(R)`$. But $`R-\wtR\in \calI_\calD`$ implies that there is polynomial $`g\in \calG_\calD`$ such that $`\lt(g)|\lt(R)`$. This contradicts the fact that $`R`$ is a normal form.

\begin{align*}
        \calG&=\{4z_1^3z_3-z_1^2z_2^2
                        -18z_1z_2z_3+4z_2^3+27z_3^2,
                        2r_1z_2^3+4z_1^2z_2z_3-z_1z_2^3
                        -54r_1z_3^2 \\
                    &~~+36z_1z_3^2-15z_2^2z_3, 6r_1z_1z_3-2r_1z_2^2-4z_1^2z_3
                        +z_1z_2^2+3z_2z_3,
                        r_1z_1z_2-9r_1z_3\\
                    &~~+6z_1z_3-2z_2^2,2r_1z_1^2-6r_1z_2-z_1z_2+9z_3,
                       3r_1^2-2r_1z_1+z_2,-z_1+2r_1+r_2\}.
\end{align*}

In the previous section, we show how to compute $`\bfmu`$-gists using Gröbner bases. This algorithm is quite slow when $`\bfmu\neq(1,1\dd 1)`$ . In the next two sections, we will introduce two methods based on an analysis of the following two $`K`$-vector spaces:

$`K^\delta\sym[\bfx]`$: the set of symmetric homogeneous polynomials of degree $`\delta`$ in $`K[\bfx]`$

$`K^\delta_\bfmu[\bfr]`$: the set of $`\bfmu`$-symmetric polynomials of degree $`\delta`$ in $`K[\bfr]`$ The first method is based on preprocessing and reduction: we first compute a basis for $`K^\delta_\bfmu[\bfr]`$, and then use the basis to reduce $`F(\bfr)`$. The second method directly computes the $`\bfmu`$-gist of $`F(\bfr)`$ by solving linear equations.

We first consider $`K^\delta\sym[\bfx]`$, the symmetric homogeneous polynomials of degree $`\delta`$. This is a $`K`$-vector space. By a of an integer $`k`$, we mean and no part $`\alpha_i`$ larger than $`n`$, we will write

\bfalpha\vdash (\delta,n).

Let

\e_\bfalpha \as \prod_{i=1}^\delta \e_{\alpha_i}

For instance if $`\delta=4, n=2, \bfalpha=(2,1,1,0)`$ then $`\e_\bfalpha=\e_2\e_1\e_1\e_0 = \e_2\e_1^2`$.

Let $`T(\bfx)`$ denote the set of terms of $`\bfx`$, and $`T^\delta(\bfx)`$ denote those terms of degree $`\delta`$. A typical element of $`T^\delta(\bfx)`$ is $`\prod_{i=1}^n x_i^{d_i}`$ where $`d_1+\cdots+d_n=\delta`$. We totally order the terms in $`T^\delta(\bfx)`$ using the lexicographic ordering in which $`x_1 \prec x_2\prec \cdots\prec x_n`$. Given any $`F\in K(\bfx)`$, its is $`\Supp(F)\ib T(\bfx)`$ such that $`F`$ can be uniquely written as F=_p(F) c(p)p where $`c:\Supp(F)\to K\setminus\set{0}`$ denote the coefficients of $`F`$. Let the $`\lt(F)`$ be equal to the $`p\in \Supp(F)`$ which is the largest under the lexicographic ordering. For instance, $`\Supp(\e_1)\,=\,\set{x_1\dd x_n}`$ and $`\lt(\e_1)\,=\,x_n`$. Also $`\Supp(\e_1\e_2) \,=\,\{x_i x_j x_k: 1\le i\neq j\le n,`$ $`1\leq k\leq n\}`$ and $`\lt(\e_1\e_2)= x_n^2x_{n-1}`$. The coefficient of $`\lt(F)`$ in $`F`$ is the of $`F`$, denoted by $`\lco(F)`$. Call $`\lm(F)\as \lco(F)\lt(F)`$ the of $`F`$. This is well-known:

The set $`\calB_1 \as \set{\e_\bfalpha: \bfalpha\vdash (\delta,n)}`$ is a $`K`$-basis for the vector space $`K^\delta\sym[\bfx]`$.

Now we consider the set $`K^\delta_\bfmu[\bfr]`$ comprising the $`\bfmu`$-symmetric functions of degree $`\delta`$. The map

\sigma_\bfmu: K^\delta\sym[\bfx] \to K^\delta_\bfmu[\bfr]

is an onto $`K`$-homomorphism. Note that $`K^\delta_\bfmu[\bfr]`$ is a vector space which is generated by the set

\mathcal{\olB}_1\as \set{\ol{G}: G\in \calB_1}

where $`\ol{G}`$ is a short hand for writing $`\sigma_\bfmu(G)`$. It follows that there is a maximal independent set $`\calB_2\ib \mathcal{\olB}_1`$ that is a basis for $`K^\delta_\bfmu[\bfr]`$. The set $`\calB_2`$ may be a proper subset of $`\mathcal{\olB}_1`$, which is seen in this example: let $`\bfmu=(2,2)`$ and $`\delta=3`$. From Example [eg:V1space], we have $`\calB_1=\set{\e_1^3, \e_1\e_2,\e_3}`$. Then

\mathcal{\olB}_1=\set{A:\ole_1^3, B:\ole_1\ole_2,
        C:\ole_3}.

We can check that $`\mathcal{\olB}_1`$ is linearly dependent since $`A+8C=4B`$. Furthermore, it is easy to verify that any $`2`$-subset of $`\mathcal{\olB}_1`$ forms a basis for $`K^\delta_\bfmu[\bfr]`$. In general, we have the following lemma.

For all $`\bfmu=(\mu_1,\mu_2)`$, $`\mathcal{\olB}_1=\{\ole_1^2, \ole_2\}`$ is a linearly independent set. Assume there exist $`k_1`$ and $`k_2`$ such that

\begin{equation}
\label{eq:s1s2linearindependent}
    k_1\ole_1^2+k_2\ole_2=0.
\end{equation}

Let $`\bfmu=(\mu_1 \dd \mu_m)`$. Then

\begin{equation}
\label{eqs:s1s2expression}
        \ole_1=\sum_{i=1}^m{\mu_ir_i},\quad
        \ole_2=\sum_{i=1}^m{\mu_i \choose 2}
            r_i^2+\sum_{i<j}{\mu_i\mu_jr_ir_j}
\end{equation}

The substitution of [eqs:s1s2expression] into [eq:s1s2linearindependent] leads to

\sum_{i=1}^m\left[k_1\mu_i^2+k_2{\mu_i\choose 2}\right]
            r_i^2+(2k_1+k_2)\sum_{i<j}\mu_i\mu_jr_ir_j=0.

Therefore,

k_1\mu_i^2+k_2{\mu_i\choose 2}=(2k_1+k_2)\mu_i\mu_j=0,
            \quad\mbox{for}\quad i,j=1 \dd m
            \quad\mbox{where}\quad i<j.

This system has a unique solution which is $`k_1=k_2=0`$. Thus it follows that $`\ole_1^2`$ and $`\ole_2`$ are linearly independent.

From the previous discussion, we saw that the dimension of $`K^\delta_\bfmu[\bfr]`$ may be smaller than that of $`K^\delta\sym[\bfx]`$. There are two special cases: when $`\bfmu=(1,1\dd 1)`$, $`dim(K^\delta\sym[\bfx])=dim(K^\delta_\bfmu[\bfr])`$; when $`\bfmu=(n)`$, $`dim(K^\delta_\bfmu[\bfr])=1`$. The following table shows the dimensions of $`K^\delta\sym[\bfx]`$ and $`K^\delta_\bfmu[\bfr]`$ for some cases. One can see that it is quite common to have a dimension drop from the specialization $`\sigma_\bfmu`$ (these lower dimensions are underlined in the table).

Dimensions of $`K^\delta\sym[\bfx]`$ and $`K^\delta_\bfmu[\bfr]`$

This subsection is devoted to generating the basis of the vector space $`K^{\delta}_{\bfmu}[\bfr]`$ with which one could easily check whether a given polynomial is in this vector space or not. For this purpose, we introduce a reduction procedure and its applications. This yields a more efficient method to check for $`\bfmu`$-symmetry .

A set $`\calB\ib K[\bfr]`$ is if any non-trivial $`K`$-linear combination over $`\calB`$ is non-zero; otherwise, $`\calB`$ is . We say $`\calC=(C_1\dd C_\ell)`$ is a if the set $`\set{C_1\dd C_\ell}`$ is linearly independent and $`\lt(C_i)\prec\lt(C_j)`$ for all $`i

We will introduce the concept of reduction. As motivation, first express any non-zero polynomial $`G`$ as $`G=\lm(G)+R`$ where $`R`$ is the tail of $`G`$ (i.e., remaining terms of $`G`$). In the terminology of term rewriting systems (e.g., and ), we then view $`G`$ as a rule for rewriting an arbitrary polynomial $`F`$ in which any occurrence of $`\lt(G)`$ in $`\Supp(F)`$ is removed by an operation of the form $`F' \ass F- c\cdot G`$, with $`c\in K`$ chosen to eliminate $`\lt(G)`$ from $`\Supp(F')`$. For instance, consider $`F= \ul{r_2^2} + 2r_1r_2 - r_1^2`$ and $`G=\ul{r_1r_2} + r_1^2 - r_2`$ where we have underlined the leading monomials of $`F`$ and $`G`$. Here we use the above convention that $`r_1\prec r_2`$. Then $`F' = F-2G = \ul{r_2^2} - 3r_1^2 + 2r_2`$. We say that $`F`$ has been reduced by $`G`$ to $`F'=F-2G`$. The $`\Supp(F')`$ no longer has $`r_1r_2`$, but has gained other terms which are smaller in the $`\prec`$-ordering.

If $`\lt(G) \notin \Supp(F)`$, we say $`F`$ is reduced relative to $`G`$. For a sequence $`\calC`$, if $`F`$ is reduced with relative to each $`G\in \calC`$, we say $`F`$ is reduced relative to $`\calC`$. Then we have this basic property:

Let $`F\neq 0`$ and $`\calC=(C_1\dd C_\ell)`$ be a canonical sequence. If $`F`$ is reduced relative to $`\calC`$, then $`\set{F,C_1\dd C_\ell}`$ is linearly independent.

By way of contradiction, assume $`F`$ is linearly dependent on $`\calC`$, say $`F=\sum_{i=1}^\ell k_iC_i`$. This implies $`\lt(F)=\lt(\sum_{i=1}^\ell k_iC_i)\preceq\lt(C_\ell)`$. So there is a smallest $`j\le\ell`$ such that $`\lt(F)\preceq\lt(C_j)`$. Since $`F`$ is reduced relative to $`\calC`$, we have $`\lt(F)\prec\lt(C_j)`$. It is easy to see that this implies $`k_j,k_{j+1}\dd k_\ell`$ are all zero. It follows that $`j\ge 2`$ (otherwise $`F=\sum_{i=1}^\ell k_iC_i=0`$). Moreover, we have $`\lt\Big(\sum_{i=1}^\ell k_iC_i\Big)\preceq \lt(C_{j-1})\prec\lt(F)`$. This contradicts the assumption $`\sum_{i=1}^\ell k_iC_i =F`$.

We next introduce the  subroutine in Figure 2 which takes an arbitrary polynomial $`F\in K[\bfr]`$ and a canonical sequence $`\calC`$ as input to produce a reduced polynomial relative to $`\calC`$.

The algorithm $`\reduce(F,\calC)`$ halts and takes at most $`\#\Supp(F)-1+\sum_{i=1}^\ell\#\Supp(C_i)`$ loops. Moreover, this bound is tight in the worst case.

Let $`F_1`$ denote the input polynomial. The variable $`F`$ in the algorithm is initially equal to $`F_1`$. In general, let $`F_j`$ ($`j=1,2,\ldots`$) be the polynomial denoted by $`F`$ at the beginning of the $`j`$th iteration of the while-loop. Thus $`p_j=\lt(F_j)`$ is the term denoted by the variable $`p`$ in the $`j`$th iteration. Note that $`F_j`$ transforms to $`F_{j+1}`$ by losing its leading term $`p_j`$ or furthermore, if $`i(j)`$ is the current value of the variable $`i`$, and $`p_j=\lt(C_{i(j)})`$ where $`C_{i(j)}\in\calC`$, we also subtract the tail of $`\frac{\lco(F_j)}{\lco(C_{i(j)})}\cdot C_{i(j)}`$ from $`F_{j+1}`$. Thus, $`\Supp(F)\ib \Supp(F_1)\cup \Supp(\calC)`$. Since $`p_1\succ p_2\succ \cdots`$ and $`p_j\in \Supp(F_1)\cup \Supp(\calC)`$, this proves that the algorithm halts after at most $`\#\Supp(F_1)+\#\Supp(\calC)`$ iterations.

Let $`L`$ be the actual number of iterations. We now give a refined argument to show that $`L\le \#\Supp(F)-1+\#\Supp(\calC)`$, i.e., we can improve the previous upper bound on $`L`$ by one. Note that we exit the while-loop when $`F=0`$ or $`i=0`$ holds. There are two cases.

CASE 1: $`F=0`$ and $`i=0`$ both hold. This implies that in the previous iteration, $`p_L = \lt(C_1)`$, and $`i`$ was decremented from $`1`$ to $`0`$. Since $`p_L`$ came from $`\#\Supp(F_1)`$ or $`\#\Supp(C_2\dd C_\ell)`$, this implies

L\le \#(\Supp(F_1)\cup \Supp(\calC))
        \le \#\Supp(F_1)-1+\#\Supp(\calC).

CASE 2: $`F\neq 0`$ or $`i>0`$. Each iteration can be “charged” to an element of $`\#(\Supp(F_1)\cup \Supp(\calC))`$. If $`i>0`$, then some elements in $`\Supp(C_1)`$ are not charged. If $`F\neq 0`$, then $`\Supp(F)\ib \Supp(F_1)\cup\Supp(\calC)`$ also implies that some elements of $`\Supp(F_1)\cup \Supp(\calC)`$ are not charged. Thus CASE 2 implies

L\le \#\Supp(F_1)-1+\#\Supp(\calC).

This proves our claimed upper bound on $`L`$.

To prove that this bound is tight, let $`F_1=p_1+q_1+\cdots+q_s`$ and $`\calC=(p_1\dd p_\ell)`$ with the term ordering $`p_1\prec\cdots\prec p_\ell\prec q_1\prec\cdots \prec q_s`$. In the first $`s`$ loops, since $`\lt(F_1)\succ p_\ell`$, $`i`$ is unchanged and $`q_1\dd q_s`$ are removed from $`F`$. In the next $`\ell-1`$ loops, since $`\lt(F_1)=p_1\prec p_2\prec\cdots\prec p_\ell`$, $`F`$ is unchanged and $`i`$ will drop to $`1`$. In the last loop, since $`\lt(F_1)=p_1=\lt(C_1)`$, $`F`$ will be reduced relative to $`C_1`$ to $`0`$. So the total number of loops is $`s+\ell=\#\Supp(F_1)-1+\sum_{i=1}^\ell \#\Supp(C_i)`$.

(Correctness) The  subroutine is correct.

Correctness of the output $`R_*`$ in the  subroutine amounts to two assertions.
(A1) The output $`R_*`$ is reduced relative to $`\calC`$.
(A2) $`F_1-R_*`$ is a linear combination of the polynomials in $`\calC`$ where $`F_1`$ is the input polynomial.
To prove these assertions, assume that the while-loop terminates after the $`L`$-th iteration. Also let $`F_j`$, $`R_j`$ and $`i_j`$ denote the values of the variables $`F`$, $`R`$ and $`i`$ at the start of the $`j`$th iteration (for $`j=1\dd L,L+1`$). Thus, $`F_1`$ is the input polynomial, $`R_1=0`$ and $`i_1=\ell`$. Assertion (A2) follows from the fact that in each iteration, the value of $`F+R`$ does not change or it changes by a scalar multiple of some $`C_i\in\calC`$. To see Assertion (A1), we use induction on $`j`$ to conclude that $`F_j`$ is reduced with respect to $`\calC_j \as (C_{1+i_j}, C_{2+i_j}\dd C_{\ell})`$, and $`R_j`$ is reduced with respect to $`\calC`$. Finally, the output $`R_*`$ is equal to $`R_{L+1}+F_{L+1}`$, At termination, there are two cases: either $`F_{L+1}=0`$ (so $`R_*=R_{L+1}`$) or $`i_{L+1}=0`$ (so $`R_*=R_{L+1}+F_{L+1}`$). In the first case, Assertion (A1) holds because $`R_*=R_{L+1}`$ and $`R_{L+1}`$ is reduced w.r.t. $`\calC`$. In the second case, Assertion (A1) holds because $`F_{L+1}`$ is reduced w.r.t. $`\calC_{L+1}=\calC`$.

If $`\calC=(C_1\dd C_\ell)`$ is canonical, then $`\reduce(F,\calC)\!=\!0`$ iff $`\set{F,C_1\dd C_\ell}`$ is linearly dependent. One direction is immediate: $`\reduce(F,\calC)=0`$ implies that $`F`$ is a linear combination of the elements of $`\calC`$. Conversely, if $`\reduce(F,\calC)=F'\neq 0`$, then $`\set{F',C_1\dd C_\ell}`$ is linearly independent by . Moreover, $`F'=F-\sum_{i=1}^\ell k'_iC_i`$ for some $`k'_1\dd k'_\ell`$. By way of contradiction, assume that $`\set{F,,C_1\dd C_\ell}`$ is linearly dependent, i.e., $`F=\sum_{i=1}^\ell k_iC_i`$ for some $`k_1\dd, k_\ell`$. It follow that $`F'=\sum_{i=1}^\ell (k_i-k'_i)C_i`$, contradicting the linear independence of $`\set{F',C_1\dd C_\ell}`$.

This gives rise to the $`\canonize`$ algorithm in Figure 3 to construct a canonical sequence.

We view the sequence $`\calC=(C_1\dd C_m)`$ as a sorted list of polynomials, with $`\lt(C_i)\prec\lt(C_{i+1})`$. Thus $`\Insert(B,\calC)`$ which inserts $`B`$ into $`\calC`$, can be implemented in $`O(\log m)`$ time with suitable data structures. The overall complexity is $`O(\ell+ m\log m)`$ where $`m`$ is the length of the output $`\calC`$. Alternatively, we could initialize the input $`\calB`$ as a priority queue can pop the polynomial $`B\in \calB`$ with the largest $`\lt(B)`$. This design yields a complexity of $`O(\ell\log \ell)`$ which is inferior when $`\ell\gg m`$.

The termination of $`\canonize(\calB)`$ is immediate from the termination of $`\reduce(F,`$ $`\calC)`$. The correctness of the output of $`\canonize(\calB)`$ comes from two facts: the returned $`\mathcal{C}`$ is clearly canonical. It is also maximal because any element $`B\in \calB`$ that does not contribute to $`\mathcal{C}`$ is clearly dependent on $`\mathcal{C}`$.

It should be pointed out that by tracking the “quotients" of $`F`$ relative to $`\calC`$ in the  algorithm and integrating the information into the  algorithm, we can derive the relationship between $`\calB=\{\ole_{\bfalpha}: \bfalpha\vdash(\delta,n)\}`$ and $`\calC=\canonize(\calB)`$ and write polynomials in $`\calC`$ as linear combinations of polynomials in $`\calB`$. By “quotients”, we mean the coefficients $`c_i`$’s in the expression $`F=\sum_{i=1}^\ell c_iC_i+R`$.

In this subsection, we use $`\reduce`$ and $`\canonize`$ algorithms to construct the $`\crgist`$ algorithm for computing the $`\bfmu`$-gist of a polynomial.

\ooF=(z_1^2,z_2)\cdot Q\cdot q^T=z_1^2-z_2.

Since termination of the algorithm $`\crgist`$ is immediate from that of $`\canonize`$ and $`\reduce`$, we only show its correctness. Assume $`\deg(F,\bfr)=\delta`$. Recall that $`F\in K[\bfr]`$ is $`\bfmu`$-symmetric iff there exists a homogeneous symmetric polynomial $`\whF\in K[\bfx]`$ of degree $`\delta`$ such that $`\sigma_{\bfmu}(\whF)=F(\bfr)`$. By , $`\whF`$ is symmetric and with degree $`\delta`$ iff $`\whF\in K^\delta\sym[\bfx]`$. Thus $`F=\sigma_{\bfmu}(\whF)\in K^\delta_\bfmu[\bfr]`$ where $`K^\delta_\bfmu[\bfr]`$ is a $`K`$-vector space with the basis generated by $`\calB=\{\ole_\bfalpha: \bfalpha\vdash(\delta,n)\}`$. If $`\calC= \canonize(\calB)`$, then $`\calC`$ is the basis we want to obtain. Therefore, if $`F`$ is $`\bfmu`$-symmetric iff $`\reduce(F,\calC)=0`$.

In this subsection, we consider an alternative reduction process where each reduction step is non-deterministic. We prove that this version can be exponential in the worst case.

For any term $`p`$, let $`\coef(F,p)`$ denote the coefficient of $`p`$ in $`F`$. If $`p\notin\Supp(F)`$, then $`\coef(F,p)=0`$. For any polynomial $`C`$, define

\reduceStep(F,C)\ass
            F-\frac{\coef(F,\lt(C))}{\lco(C)} C.

We call $`\reduceStep(F,C)`$ a or a $`\calC`$-reduction step in case $`C\in \calC`$. We see that $`\reduceStep(F,C)=F`$ iff $`\lt(C)`$ does not occur in $`F`$. We say the reduction is improper in this case.

Let $`\nreduce(F,\calC)`$ denote the subroutine that repeatedly transforms $`F`$ by applying proper $`\calC`$-reduction steps to $`F`$ until no more more change is possible. It returns the final value of $`F`$. We call this the of $`F`$.

For any linearly independent set $`\calC`$, we have

\nreduce(F,\calC)=\reduce(F,\calC).

Then $`\nreduce(F,\calC)`$ has $`\leq 2^\ell`$ $`\calC`$-reduction steps where $`\ell=|\calC|`$. Moreover, $`2^\ell`$ steps may be needed.

Let $`R_1=\nreduce(F,\calC)`$ and $`R_2=\reduce(F,\calC)`$. Then there exists $`k_1\dd k_\ell`$ and $`k_1'\dd k_\ell'`$ such that

F=\sum_{i=1}^\ell k_iC_i+R_1=\sum_{i=1}^\ell k_i'C_i+R_2.

It is immediate that

R_1-R_2=\sum_{i=1}^\ell (k_i-k_i')C_i.

If $`R_1\neq R_2`$, there exists $`i`$ such that $`k_i\neq k_i'`$ and $`k_j=k_j'~(j=1\dd i-1)`$. Then $`\lt(R_1-R_2)=\lt(C_i)`$. This implies that $`\lt(C_i)\in \Supp(R_1)`$ or $`\lt(C_i)\in\Supp(R_2)`$. Hence $`R_1`$ or $`R_2`$ is not reduced relative to $`\calC`$. This contradicts with the output requirements of $`\reduce`$ or $`\nreduce`$.

Let us define $`a_\ell`$ to be the longest $`\calC`$-derivation for any $`\calC`$ with $`\ell`$ elements. CLAIM A: $`a_\ell\le 2^\ell-1`$. Let $`\calC_\ell=(C_1\dd C_\ell)`$ be any canonical sequence with $`\ell`$ elements. Let F_0F_1F_N be any $`\calC_\ell`$-derivation. We must prove that $`N\le 2^\ell-1`$ by induction of $`\ell`$. Clearly, if $`\ell=1`$, then $`a_1\le 1=2^1-1`$. Next, inductively assume that $`a_{\ell-1}\le 2^{\ell-1}-1`$. Suppose there does not exist an $`i

The last assertion of our proposition amounts to CLAIM B: $`a_\ell\ge 2^\ell-1`$. To show this claim, let $`\calC_\ell=(C_1\dd C_\ell)`$ as before. But we now choose $`C_i \as \sum_{j=1}^i p_j`$ where $`p_j`$’s are terms satisfying $`p_j\prec p_{j+1}`$. Let us write

F \xrightarrow[ k ]{ \calC } G

to mean that there is a $`\calC`$-derivation of length $`k`$ from $`F`$ to $`G`$. Our claim follows if we show that

C_\ell \xrightarrow[ 2^\ell-1 ]{ \calC_\ell } 0.

The basis is obvious: $`C_1 \xrightarrow[ 1 ]{ \calC_l } 0.`$ Inductively, assume that C_ 0. The inductive assumption implies

C_{\ell} = p_{\ell}+C_{\ell-1}
            \xrightarrow[ 2^{\ell-1}-1 ]{ \calC_{\ell-1}}
            p_{\ell}.

Next, in one step, we have $`p_{\ell} \xrightarrow[ 1 ]{ \calC_{\ell} } -C_{\ell-1}`$ and, again from the induction hypothesis,

-C_{\ell-1}
            \xrightarrow[ 2^{\ell-1}-1 ]{ \calC_{\ell-1} } 0.

Concatenating these 3 derivations, shows that $`C_\ell\xrightarrow[ 2^\ell-1] {\calC_\ell} 0`$. This proves CLAIM B.

In this section, we introduce a direct method to compute gist of $`F(\bfr)`$ without preprocessing. Such methods depend on the choice of basis for $`K^\delta\sym[\bfr]`$. Our default basis is elementary symmetric polynomials.

Our algorithm that takes as input $`F\in K[\bfr]`$ and $`\bfmu`$, and either outputs the $`\bfmu`$-gist $`\ooF`$ of $`F`$ or detects that $`F`$ is not $`\bfmu`$-symmetric. The idea is this: $`F`$ is $`\bfmu`$-symmetric iff $`\ooF`$ exists. The existence of $`\ooF`$ is equivalent to the existence of a solution to a linear system of equations. More precisely, there is an polynomial identity of the form $`\ooF(\ole_1\dd \ole_n)=F.`$ To turn this identity into a system of linear equations, we first construct a polynomial

G(\bfk;\bfz)\in K[\bfk][\bfz]

in $`\bfz`$ with indeterminate coefficients in $`\bfk`$, with homogeneous weighted degree $`\delta`$ in $`\bfz`$ (see Section 2.1 for definition of weighted degree). Here $`\delta`$ is the degree of $`F`$. Each term is of weighted degree $`\delta`$ and has the form

\bfz_\bfalpha \as \prod_{i=1}^\delta z_{\alpha_i}

where $`\bfalpha=(\alpha_1\dd \alpha_\delta)`$ is a weak partition of $`\delta`$ with parts at most $`n`$, i.e., $`\bfalpha\vdash (\delta,n)`$. Next, we plug in $`\ole_i`$’s for the $`z_i`$’s to get

H(\bfk;\bfr)\as G(\bfk;\ole_1\dd \ole_n)

viewed as a polynomial in $`K[\bfk][\bfr].`$ We then set up the equation H(;) = F() to solve for the values of $`\bfk`$. Note that total degree of $`G`$ in $`\bfk`$ is $`1`$, i.e., $`\deg(G,\bfk)=1`$. Therefore, $`\deg(H,\bfk)=1`$. Thus amounts to solving a linear system of equations in $`\bfk`$.

To illustrate this process, consider the polynomial $`F=3r_1^2+2r_1r_2+r_2^2`$ and $`\bfmu=(2,1)`$.

Assign $`\delta= \deg(F,\bfr)=2`$ and $`n= \sum_{i=1}^m\mu_i=3`$.

the terms of weighted degree $`2`$ are $`z_1^2`$ and $`z_2`$.

Construct the polynomial $`G(\bfk;\bfz)\as k_1z_1^2+k_2z_2`$ where $`\bfk=(k_1,k_2)`$ are the indeterminate coefficients.

Using $`\ole_1=2r_1+r_2, \ole_2=r_1^2+2r_1r_2`$, construct the polynomial

H(\bfk;\bfr)\as G(\bfk;\ole_1\dd\ole_n)=
        (4k_1+k_2)r_1^2 + (4k_1+2k_2)r_1r_2 + k_1r_2^2.

Extract the coefficient vector $`\coeffs(H,\bfr)`$ of $`H(\bfk;\bfr)`$ viewed as a polynomial in $`\bfr`$. The entries of this vector are linear in $`\bfk`$. Thus $`H= \coeffs(H,\bfr)\cdot T^\delta(\bfr)`$ where $`T^\delta(\bfr)`$ is the vector of all terms of $`T(\bfr)`$ of degree $`\delta`$.

Extract the coefficient vector $`\coeffs(F,\bfr)`$ of $`F(\bfr)`$. This vector is a constant $`(3,2,1)^T`$ .

The last two steps enables the construction of a system of linear equations, $`A\bfk =\bfb`$: H(;) &=& F()
(4k_1+k_2)r_1^2 + (4k_1+2k_2)r_1r_2 + k_1r_2^2 &=& 3r_1^2+2r_1r_2+r_2^2
(H,) &=& (F,)
&=&
A &=& where the last equation is the linear system to be solved for $`\bfk=\mmat{k_1\\k_2}`$.

If $`A \bfk=\bfb`$ has no solutions, we conclude that $`F`$ is not $`\bfmu`$-symmetric. Otherwise, choose any solution for $`\bfk`$ and plugging into $`G(\bfk;\bfr)`$, we obtain a gist of $`F(\bfr)`$. Note that there may be multiple solutions for $`\bfk`$ because of the presence of $`\bfmu`$-constraints.

We now summarize the above procedure as the  algorithm:

The correctness of the algorithm $`\lsgist`$ lies in the fact that $`F`$ is $`\bfmu`$-symmetric iff $`F\in K^\delta_\bfmu[\bfr]`$ which is generated by $`\{\ole_{\bfalpha}: \bfalpha\vdash(\delta,n)\}`$.

In this section, We briefly sketch how to extend the above methods to computing gists relative to other bases of $`K^\delta\sym[\bfx]`$.

The set $`K\sym[\bfx]`$ of symmetric functions can be viewed as a $`K`$-algebra generated by some finite set $`\calG`$. The following are three well-known choices of $`\calG`$ with $`n`$ elements each:

(Elementary symmetric polynomials) $`\calG_e\as \set{\e_1\dd\e_n}`$ where $`e_i`$ is the $`i`$-th elementary symmetric function of $`\bfx`$.

(Power-sum symmetric polynomials) $`\calG_p\as \set{p_1\dd p_n}`$ where $`p_i=x_1^i+\cdots+x_n^i`$.

(Complete homogeneous symmetric polynomials) $`\calG_c\as \set{c_1\dd c_n}`$ where $`c_i`$ is the sum of all distinct monomials of degree $`i`$ in the variables $`x_1\dd x_n`$.

For each $`\delta\ge 1`$, the vector space $`K\sym^\delta[\bfx]`$ of symmetric polynomials of degree $`\delta`$ has a basis $`\calB^\delta`$ that corresponds to a given generator set $`\calG`$.

The following are bases of $`K\sym^\delta[\bfx]`$:

() $`\calB_e^\delta\as \set{e_{\bfalpha}: \bfalpha\vdash(\delta,n)}`$ where $`e_{\bfalpha}=\prod_{i=1}^\delta e_{\alpha_i}`$ and $`\bfalpha=(\alpha_1\dd\alpha_\delta)`$;

() $`\calB_p^\delta\as \set{p_{\bfalpha}: \bfalpha\vdash(\delta,n)}`$ where $`p_{\bfalpha}=\prod_{i=1}^\delta p_{\alpha_i}`$;

() $`\calB_c^\delta\as \set{c_{\bfalpha}: \bfalpha\vdash(\delta,n)}`$ where $`c_{\bfalpha}=\prod_{i=1}^\delta c_{\alpha_i}`$. But $`K\sym^{\delta}[\bfx]`$ can also be generated with monomial symmetric polynomials. In this case, exactly $`n`$ parts: $`\alpha_1\ge\cdots\ge\alpha_n\ge 0`$. We also write $`\bfx^\bfalpha`$ for the product $`\prod_{i=1}^n x_i^{\alpha_i}`$. This yields yet another basis for $`K\sym^\delta[\bfx]`$:

() $`\calB_m^\delta\as \set{m_{\bfalpha}: \bfalpha\vdash(\delta)_n}`$ where $`m_{\bfalpha}=\sum_{\bfbeta} \bfx^\bfbeta`$ where $`\bfbeta`$ ranges over all permutations of $`\bfalpha`$ which are distinct. For instance, if $`\bfalpha=(2,0,0)`$ then $`\bfbeta`$ ranges over the set $`\set{(2,0,0), (0,2,0),(0,0,2)}`$ and $`m_\bfalpha=x_1^2+x_2^2+x_3^2`$.

So far, this paper has focused on the $`e`$-basis. But concepts and algorithms When using $`p`$-basis or $`c`$-basis, we only need to replace $`\ole_i`$ used by the algorithms $`\ggist`$, $`\crgist`$ and $`\lsgist`$ by $`\olp_i\as\sigma_{\bfmu}(p_i)`$ or $`\olc_i\as\sigma_{\bfmu}(c_i)`$, respectively; when using the $`m`$-basis, the index set $`\bfalpha\vdash (\delta,n)`$ should be replaced by $`\bfalpha\vdash (\delta)_n`$ and $`\ole_i`$ should be replaced by $`\olm_\bfalpha\as\sigma_{\bfmu}(m_\bfalpha)`$. The relative performance of the algorithms using different bases will be evaluated in Section 8.

In this section, we report some experimental results to show the effectiveness and efficiency of the two approaches presented in this paper. These experiments were performed using Maple on a Windows laptop with an Intel(R) Core(TM) i7-7660U CPU in 2.50GHz and 8GB RAM.

In , we compare the performance of the three algorithms described in this paper for checking the $`\bfmu`$-symmetry of polynomials: ,  and . We use a test suite of $`12`$ polynomials of degrees ranging from $`6`$–$`20`$ (see ), with corresponding $`\bfmu`$ with $`n=|\bfmu|`$ ranging from $`4`$–$`6`$. These polynomials are either $`\dplus`$ polynomials or subdiscriminants, or some perturbations (to create non-$`\bfmu`$-symmetric polynomials).

From Table 1, it is clear that  is significantly faster than . There is one anomaly in the table: for the polynomial $`F_3`$,  is 5 times faster than . This is when $`\bfmu`$ is $`(1\dd 1)`$, which indicates that the ideal $`\calI_\bfmu=\left`$ has a symmetric structure in $`\bfr`$. We believe it is because the Gröbner basis of $`\calI`$ can be computed very efficiently for certain types of structures.

From Table 2, we observe that the algorithm  is more efficient than the other algorithms in general because it doesn’t require polynomial expansion and thus can save a lot of time, especially when $`\delta`$ is big (see F3, F4 and F7). The algorithm  also behaves well because power-sum symmetric polynomials have fewer terms than elementary symmetric polynomials and complete homogeneous symmetric ones and this property may help save time during polynomial expansion. Overall, algorithms based on + are not as competitive as those based on linear system solving because the preprocessing procedure  charges more time in order to generate a canonical sequence.

However, from Table 3, we see that for fixed $`\bfmu`$ and $`\delta`$, once we have computed the canonical set in the preprocessing step, the time cost for $`\reduce`$ is small. Although the Gröbner basis method also contains a preprocessing procedure, the time cost for computing normal forms is quite expensive and thus it is not as competitive as algorithms based on + and linear system solving. Furthermore, for algorithms using $`p`$-basis, the algorithm  shows higher efficiency than  and , especially for big $`\delta`$ and $`n`$ (See F3, F4 and F7). This could be attributed to the small number of terms in the generator polynomials. In contrast, for $`e`$-basis and $`c`$-basis, the algorithms  and  prevail over  and . The possible reason might be that many terms will get canceled when computing a canonical sequence.

We have introduced the concept of $`\bfmu`$-symmetric polynomial which generalizes the classical symmetric polynomial. Such $`\bfmu`$-symmetric functions of the roots of a polynomial can be written as a rational function in its coefficients. Our original motivation was to study a conjecture that a certain polynomial $`\dplus(\bfmu)`$ is $`\bfmu`$-symmetric.

F1 & 12 & [1, 1, 1, 1] & 4 & Y & 0.453 & 0.235 & 1.9 & 0.094 & 0.000 & 0.094 & 4.8
F2 & 8 & [2, 1, 1] & 4 & Y & 0.328 & 0.015 & 21.9 & 0.016 & 0.015 & 0.031 & 10.6
F3 & 20 & [1, 1, 1, 1, 1] & 5 & Y & 34.1 & 188 & 0.2 & 3.77 & 0.031 & 3.80 & 9.0
F4 & 15 & [2, 1, 1, 1] & 5 & Y & $`>`$600 & 1.88 & $`>`$320 & 0.391 & 0.015 & 0.406 & $`>`$1478
F4x & 6 & [2, 1, 1, 1] & 5 & N & $`>`$600 & 0.015 & $`>`$4$`\times\!10^4`$ & 0.000 & 0.016 & 0.016 & $`>`$3.7$`\times\!10^4`$
F5 & 6 & [2, 2, 1] & 5 & Y & 68.0 & 0.032 & 2126 & 0.000 & 0.000 & 0.000 & Inf
F5x & 6 & [2, 2, 1] & 5 & N & 0.078 & 0.000 & Inf & 0.000 & 0.016 & 0.016 & 4.9
F6 & 10 & [2, 2, 1] & 5 & Y & 0.438 & 0.078 & 5.6 & 0.031 & 0.000 & 0.031 & 14.1
F6x & 10 & [2, 2, 1] & 5 & N & 0.406 & 0.047 & 8.6 & 0.031 & 0.016 & 0.047 & 8.6
F7 & 18 & [3, 1, 1, 1] & 6 & Y & $`>`$600 & 9.00 & $`>`$66.7 & 3.39 & 0.063 & 3.45 & $`>`$174
F8 & 12 & [3, 2, 1] & 6 & Y & $`>`$600 & 0.360 & $`>`$1667 & 0.187 & 0.000 & 0.187 & $`>`$3210
F9 & 6 & [2, 2, 2] & 6 & Y & 8.73 & 0.000 & Inf & 0.000 & 0.000 & 0.000 & Inf

F1 & 0.219 & 0.344 & 0.187 & 0.063 & 0.187 & 0.078 & 0.094 & 0.297 & 0.094 & 0.500 & 0.031
F2 & 0.328 & 387 & 565 & 0.015 & 1.00 & 0.032 & 0.015 & 0.016 & 0.000& 0.031 & 0.000
F3 & 20.9 & 61.2 & 60.3 & 3.19 & 313 & 14.7 & 2.17 & 79.3 & 3.11 & 2109 & 0.391
F4 & $`>`$3000 & $`>`$3000 & $`>`$3000 & 0.422 & 5.06 & 1.58 & 0.437 & 0.907 & 0.281 & 5.19 & 0.110
F4x & $`>`$3000 & $`>`$3000 & $`>`$3000 & 0.000& 0.015 & 0.016 & 0.016 & 0.015 & 0.016 & 0.031 & 0.000
F5 & 41.2 & $`>`$3000 & $`>`$3000 & 0.016 & 0.016 & 0.015 & 0.016 & 0.000& 0.015 & 0.016 & 0.000
F5x & 0.047 & $`>`$3000 & $`>`$3000 & 0.015 & 0.016 & 0.015 & 0.016 & 0.000& 0.016 & 0.015 & 0.000
F6 & 0.234 & $`>`$3000 & $`>`$3000 & 0.047 & 0.094 & 0.093 & 0.063 & 0.031& 0.031& 0.063 & 0.032
F6x & 0.281 & $`>`$3000 & $`>`$3000 & 0.047 & 0.094 & 0.093 & 0.063 & 0.031& 0.031& 0.047 & 0.031
F7 & $`>`$3000 & $`>`$3000 & $`>`$3000 & 3.50 & 63.3 & 29.8 & 3.50 & 5.63 & 1.70 & 49.2 & 1.52
F8 & $`>`$3000 & $`>`$3000 & $`>`$3000 & 0.234 & 0.641 & 0.656 & 0.438 & 0.156 & 0.109& 0.250 & 0.157
F9 & 6.17 & $`>`$3000 & $`>`$3000 & 0.016 & 0.000& 0.031 & 0.016 & 0.000& 0.015 & 0.031 & 0.016