4
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$\newcommand{QQ}{\mathbb{Q}}$

Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial

$$ f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i} $$

Now let $$ g_j = \mathrm{resultant}_x(f, \frac{d^{m+1-j}}{dx^{m+1-j}} f) $$ for $j=1,\ldots,m$ and let $$ I_s = (g_1,\ldots, g_s) $$ be an ideal of $R$ (and of $S=\QQ[a_1,\ldots,a_m]$).

Now if one computes a gröbner base of $I_s$ in $R$ or $S$ with respect to the natural graded reverse lexicographic monomial order (the default order in Macaulay2) one finds experimentally, that the radical of the monomial ideal generated by the leading monomials of the gröbner base (which is identical to the radical of the leading monomials of $I_s$ itself, by gröbner base theory) is of the form $$ \sqrt{\mathrm{lmon}(I_s)} = (a_1,\ldots,a_s) $$

I checked this for $m+1 = 2, 3, 4, 5$ with the following little piece of Macaulay2 code

compresulseq = (nn) ->
(

    m := nn - 1;

    R := QQ[x,a_1..a_m];

    f := x^nn + sum toList apply(1..m, jj->a_jj^jj * x^(nn - jj));

    resulseq := toList apply(1..m, jj->resultant(f, diff(x^(nn - jj), f), x));

    resulseq

);

getinitterms2 = (reslissecond) ->
(

    m := length reslissecond;

    result := toList apply(1..m, jj->ideal (take ( reslissecond, {0, jj-1})));

    result = apply(result, ww-> radical ideal apply(flatten entries gens gb ww, uu->leadTerm uu));

    result

);

Has anyone an idea how one could prove this or what could be helpful literature for further research on this question?

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