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Implicitization by symmetric polynomials

asked 2020-05-09 04:16:22 -0500

A.P. gravatar image

updated 2020-05-09 07:18:04 -0500

Let $p_1,\dotsc,p_m$ be real polynomials (although rational can do as well) in $t$ variables.

What is the most efficient way to compute the smallest degree $d$ such that $s(p_1,\dotsc,p_m) = 0$ for some rational symmetric polynomial $s$ in $m$ variables?


At the moment I am doing this for $t = 1$ by iterating the following simple algorithm over $d \geq 1$:

  1. Create a list Sc of the Schur functions corresponding to the partitions of $k \leq d$ with at most $m$ parts, which is a linear basis over $\Bbb{Q}$ for the space of symmetric polynomials in $m$ variables with degree up to $d$.

  2. Compute the list Ps of the evaluations of each element of Sc at $(p_1,\dotsc,p_m)$.

  3. Convert the elements of Ps into vectors and stop if they are linearly dependent, otherwise increase $d$ by one and repeat from 1.

This works reasonably well when the degrees of $p_1,\dotsc,p_m$ are small, but otherwise iterating over each $d$ involves quite a bit of work and it doesn't scale very well.

I know that the equivalent problem for $s \in \Bbb{Q}[X_1,\dotsc,X_m]$ is "readily" solved through variable elimination via Gröbner bases, but I have yet to find a way to make this work for symmetric polynomials. I also thought that I might compute the relevant ideal and then try to find the subset fixed by the symmetric group in $m$ variables, but I couldn't find any facilities in Sage to compute fixed sets under a group action (which I guess is a hard problem in general).


This is my current code:

S = SymmetricFunctions(QQ).s()
R.<x> = QQ[]

def eval_schur(F, d):
    """
    Returns the evaluation at a vector F of the Schur functions of degree n.
    """
    m = len(F)
    schur_polys = ((l, S(l).expand(m)) for l in Partitions(d) if len(l) <= m)
    return ((l, sch(*F)) for (l,sch) in schur_polys)

def poly_vectors(polys):
    d = max(p.degree() for p in polys)
    return (p.padded_list(d+1) for p in polys)

def order_of_symmetric_dependence(F):
    schur = [R(1)]
    vectors = [1]
    d = 0
    while not FreeModule(QQ,1).are_linearly_dependent(vectors):
        d += 1
        schur.extend((s for (_,s) in eval_schur(F,d)))
        vectors = poly_vectors(schur)
    return d

And here's a sample output:

In [2]: [order_of_symmetric_dependence([x, x^2, x^(2*k)]) for k in range(1,8)]
Out[2]: [4, 5, 8, 11, 12, 12, 12]
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Comments

Can you add your code and a sample case? Maybe naive idea: eliminate $x_1,\ldots,x_t$ from $\langle y_i - e_i(p_1(x_1,\ldots,x_t),\ldots,p_m(x_1,\ldots,x_t)) \rangle$ where $e_i$ are the elementary symmetric polynomials in $m$ variables.

rburing gravatar imagerburing ( 2020-05-09 04:53:41 -0500 )edit
1

@rburing I had to clean up my code a little, but here it is.

A.P. gravatar imageA.P. ( 2020-05-09 07:19:23 -0500 )edit

@rburing Unfortunately, your suggestion doesn't seem to work, assuming I understand it correctly:

F = [t,t^2,t^3]
Rt.<t,x1,x2,x3> = QQ[]
I = Rt.ideal([x1 - E[1].expand(3)(F), x2 - E[2].expand(3)(F), x3 - E[3].expand(3)(F)])
gs = I.elimination_ideal(t).gens()
[g(x1 = F[0], x2 = F[1], x3 = F[2]) for g in gs]

outputs

[-t^6 - t^5 + t^4 + t^3,
t^7 - t^6 - 3*t^5 + t^4 + 2*t^3,
t^5 + t^4 - t^3 - t^2,
2*t^10 - 4*t^9 - 9*t^8 + 2*t^7 + 5*t^6]

instead of [0,0,0,0].

A.P. gravatar imageA.P. ( 2020-05-09 07:37:08 -0500 )edit
1

Actually, I made a mistake there. I should have substituted $e_i$ back in after eliminating the variables, and indeed evaluating the elements of

gs = [g(x1 = E[1].expand(3), x2 = E[2].expand(3), x3 = E[3].expand(3)) for g in I.elimination_ideal(t).gens()]

at F results in [0,0,0,0]. I'll have to do some performance tests, now.

A.P. gravatar imageA.P. ( 2020-05-09 07:52:05 -0500 )edit

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answered 2020-05-09 12:34:31 -0500

rburing gravatar image

As suggested in my comment: you can eliminate $x_1,…,x_t$ from the ideal $$I = \langle y_i - e_i(p_1(x_1,\ldots,x_t),\ldots,p_m(x_1,\ldots,x_t)) : i = 1,\ldots, m \rangle$$ in the ring $\mathbb{Q}[x_1,\ldots,x_t,y_1,\ldots,y_m]$, where $e_i$ are the elementary symmetric polynomials in m variables.

For efficiency reasons we compute the elimination ideal $I \cap \mathbb{Q}[y_1,\ldots,y_m]$ manually, using a Groebner basis $G$ of $I$ with respect to a block ordering where the $x$'s are the greatest and the $y$'s have a weighted degrevlex ordering where $y_k$ has degree $k$ (simulating the degree of $e_k$), so that substituting $y_i = e_i(x_1,\ldots,x_t)$ into the polynomials in $G \cap \mathbb{Q}[y_1,\ldots,y_m]$ yields symmetric dependencies of minimal degree.

def symmetric_dependencies(p):
    R = p[0].parent()
    m = len(p)
    S = PolynomialRing(R.base_ring(),
                       names=list(R.gens()) + ['y{}'.format(k+1) for k in range(m)],
                       order=TermOrder('degrevlex', R.ngens()) + \
                             TermOrder('wdegrevlex',list(range(1,m+1))))
    p = [S(f) for f in p]
    X = S.gens()[0:R.ngens()]
    Y = S.gens()[R.ngens():]
    E = SymmetricFunctions(QQ).elementary()
    Es = [E[k+1].expand(m) for k in range(m)]
    I = S.ideal([Y[k] - Es[k](p) for k in range(m)])
    G = I.groebner_basis()
    GY = [g for g in G if all(v in Y for v in g.lt().variables())]
    E_subs = {Y[k] : Es[k] for k in range(m)}
    dependencies = [f.subs(E_subs) for f in GY]
    #assert all(f(p) == 0 for f in dependencies)
    return dependencies

Example:

sage: R.<x> = QQ[]
sage: [min(f.degree() for f in symmetric_dependencies([x, x^2, x^(2*k)])) for k in range(1,8)]
[4, 5, 8, 11, 12, 12, 12]

It seems to be slightly faster than your function on this example.

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1

Thank you, this is precisely what I was after.

A.P. gravatar imageA.P. ( 2020-05-09 16:15:18 -0500 )edit

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Asked: 2020-05-09 04:16:22 -0500

Seen: 109 times

Last updated: May 09