History of Segre classes $s_n(c_1,\dots,c_n)$ in terms of Chern classes

   comment-component: The table stores one homogeneous component per row, not the total     Segre class $1+s_1+s_2+\cdots$ truncated at degree $n$.-  comment-zero: The constant component is $s_0=1$ and is stated in Formula CITE{formula-zero},-    rather than stored as a row.+  comment-zero: The constant component is $s_0=1$ and is not stored as a row; the+    table begins at $n=1$.   comment-rank: There is no rank parameter in the table. For a vector bundle of rank     $r$, the universal polynomial is specialised by setting $c_j=0$ for $j>r$.

     h_n(x_1,x_2,\dots)$, where $h_n$ is the HREF{Complete_homogeneous_symmetric_polynomials}[complete     homogeneous symmetric polynomial].-  formula-zero: $s_0=1$.-  formula-first: $s_1=-c_1$, $s_2=c_1^2-c_2$, and $s_3=-c_1^3+2c_1c_2-c_3$. Programs:   program-sage:

       \  R = PolynomialRing(ZZ, [\"c%s\" % i for i in range(1, n + 1)])\n    c = [R(1)]\       \ + list(R.gens())\n    s = [R(1)]\n    for m in range(1, n + 1):\n        s.append(-sum(c[j]\-      \ * s[m - j] for j in range(1, m + 1)))\n    return s[n]\n\nprint(segre_component(6))"+      \ * s[m - j] for j in range(1, m + 1)))\n    return s[n]\n\nprint(segre_component(7))" Similar tables: - table: HREF{Todd_polynomials}[Todd polynomials]

     variables - table: HREF{Chern_character_polynomials}[Chern character polynomials]-  relation: store another universal characteristic-class polynomial in the same Chern-class-    variables+  relation: store the components of $\operatorname{ch}(E)=\sum_i e^{x_i}$ in the same+    Chern-class variables - table: HREF{Complete_homogeneous_symmetric_polynomials}[Complete homogeneous symmetric     polynomials]-  relation: give the $h_n$ whose signed Chern-root expressions are stored here+  relation: give the $h_n$ in the Chern roots; the entries here are $(-1)^n h_n$ rewritten+    in the Chern classes - table: HREF{Elementary_symmetric_polynomials}[Elementary symmetric polynomials]   relation: give the elementary symmetric functions $e_j$ that are renamed as Chern

     the Chern-root identity in Formula CITE{formula-roots} and by substituting the     Chern classes of $T\mathbb P^m$ for $1\leq n\leq m\leq 6$, which gives $s_n(T\mathbb-    P^m)=(-1)^n\binom{m+n}{n}h^n$.+    P^m)=(-1)^n\binom{m+n}{n}h^n$, where $h$ is the hyperplane class. Display properties:   number-header: $s_n$ Numbers:-- params:-    n: '1'-  number: -c1-- params:-    n: '2'-  number: c1^2 - c2-- params:-    n: '3'-  number: -c1^3 + 2*c1*c2 - c3-- params:-    n: '4'-  number: c1^4 - 3*c1^2*c2 + c2^2 + 2*c1*c3 - c4-- params:-    n: '5'-  number: -c1^5 + 4*c1^3*c2 - 3*c1*c2^2 - 3*c1^2*c3 + 2*c2*c3 + 2*c1*c4 - c5-- params:-    n: '6'-  number: c1^6 - 5*c1^4*c2 + 6*c1^2*c2^2 + 4*c1^3*c3 - c2^3 - 6*c1*c2*c3 - 3*c1^2*c4+  '1': -c1+  '2': c1^2 - c2+  '3': -c1^3 + 2*c1*c2 - c3+  '4': c1^4 - 3*c1^2*c2 + c2^2 + 2*c1*c3 - c4+  '5': -c1^5 + 4*c1^3*c2 - 3*c1*c2^2 - 3*c1^2*c3 + 2*c2*c3 + 2*c1*c4 - c5+  '6': c1^6 - 5*c1^4*c2 + 6*c1^2*c2^2 + 4*c1^3*c3 - c2^3 - 6*c1*c2*c3 - 3*c1^2*c4     + c3^2 + 2*c2*c4 + 2*c1*c5 - c6