History of Todd polynomials $\mathrm{Td}_n(c_1,\dots,c_n)$

   comment-component: The table stores one homogeneous component per row, not the total     Todd class $1+\mathrm{Td}_1+\mathrm{Td}_2+\cdots$ truncated at degree $n$.-  comment-zero: The constant component is $\mathrm{Td}_0=1$ and is stated in Formula-    CITE{formula-zero}, rather than stored as a row.-  comment-variables: The variables are Chern classes. Some sources write the elementary-    symmetric functions of the Chern roots as $p_i$; this table writes them as $c_i$,-    reserving $p_i$ for Pontryagin classes in the characteristic-class tables.-  comment-odd: For odd $n>1$, the coefficient of $c_n$ vanishes. Thus $\mathrm{Td}_7$-    has no $c_7$ term and still fits in the six-variable polynomial search.+  comment-zero: The constant component $\mathrm{Td}_0=1$ is not stored as a row; the+    table begins at $n=1$.+  comment-variables: The variables are Chern classes. Wikipedia writes the Todd class+    as $\operatorname{td}$ and its components as $\operatorname{td}_j$ CITE{ToddWiki}.+    Both sources cited here write the elementary symmetric functions as $p_i$ CITE{ToddWiki}+    CITE{GenusWiki}; CITE{GenusWiki} notes that the same notation is often used for+    Pontryagin classes. This table writes the class as $\mathrm{Td}$ and the variables+    as $c_i$.+  comment-odd: The coefficient of $c_n$ in $\mathrm{Td}_n$ is $B_n^+/n!$, which vanishes+    for odd $n>1$; $\mathrm{Td}_7$ therefore has no $c_7$ term. Formulas:   formula-component: $1+\sum_{n\geq1}\mathrm{Td}_n(c_1,c_2,\dots)=\prod_i Q(x_i)$,-    where $c_j=e_j(x_1,x_2,\dots)$.+    where the $x_i$ are the Chern roots and $c_j=e_j(x_1,x_2,\dots)$.   formula-series: $Q(x)=\frac{x}{1-e^{-x}}=1+\frac{x}{2}+\frac{x^2}{12}-\frac{x^4}{720}+     \frac{x^6}{30240}+\cdots$ CITE{GenusWiki}. Equivalently $Q(x)=\sum_i B_i^+x^i/i!$,     where $B_1^+=+\frac12$ and $B_i^+=B_i$ for $i\ne1$, using HREF{Bernoulli_numbers}[the     Bernoulli numbers].-  formula-zero: $\mathrm{Td}_0=1$.+Programs:+  program-sage:+    language: Sage+    code: "import numberdb.sage as numberdb  # initialize Sage before named imports\n\+      from math import factorial, prod\nfrom sage.rings.polynomial.polynomial_ring_constructor\+      \ import PolynomialRing\nfrom sage.rings.rational_field import QQ\n\ndef bernoulli_plus(n):\n\+      \    values = [QQ(0)] * (n + 1)\n    for m in range(n + 1):\n        values[m]\+      \ = QQ(1) / QQ(m + 1)\n        for j in range(m, 0, -1):\n            values[j\+      \ - 1] = QQ(j) * (values[j - 1] - values[j])\n    return values[0]\n\ndef log_q_coefficient(m):\n\+      \    if m == 1:\n        return QQ(1) / QQ(2)\n    if m % 2:\n        return\+      \ QQ(0)\n    return -bernoulli_plus(m) / QQ(m * factorial(m))\n\ndef weight(exponents):\n\+      \    return sum((i + 1) * exponent for i, exponent in enumerate(exponents))\n\+      \ndef monomial(parent, variables, exponents):\n    return parent(prod(variables[i]\+      \ ** exponent\n                       for i, exponent in enumerate(exponents)))\n\+      \ndef weighted_terms(polynomial, degree, exact):\n    parent = polynomial.parent()\n\+      \    variables = parent.gens()\n    total = parent(0)\n    for exponents, coeff\+      \ in polynomial.dict().items():\n        term_weight = weight(exponents)\n \+      \       if (exact and term_weight == degree) or (not exact and term_weight <=\+      \ degree):\n            total += coeff * monomial(parent, variables, exponents)\n\+      \    return total\n\ndef todd_component(n):\n    R = PolynomialRing(QQ, [\"\+      c%s\" % i for i in range(1, n + 1)])\n    c = R.gens()\n    power_sums = {}\n\+      \    for m in range(1, n + 1):\n        p_m = sum(((-1) ** (i + 1) * c[i - 1]\+      \ * power_sums[m - i]\n                   for i in range(1, m)), R(0))\n   \+      \     power_sums[m] = p_m + (-1) ** (m + 1) * m * c[m - 1]\n\n    exponent =\+      \ sum((log_q_coefficient(m) * power_sums[m]\n                    for m in range(1,\+      \ n + 1)), R(0))\n    total = R(1)\n    term = R(1)\n    for k in range(1, n\+      \ + 1):\n        term = weighted_terms(term * exponent / QQ(k), n, False)\n\+      \        total = weighted_terms(total + term, n, False)\n    return weighted_terms(total,\+      \ n, True)\n\nprint(todd_component(8))" Similar tables: - table: HREF{Elementary_symmetric_polynomials}[Elementary symmetric polynomials]

 - characteristic class Todd polynomial - multiplicative sequence Todd genus+- td polynomials Tags: - characteristic classes

   rigour: exact   complete: 'no'-  complete-note: it holds every entry with $1\leq n\leq 7$, and $\mathrm{Td}_7$ is-    the last entry whose polynomial uses at most six variables-  rigour details: The generator computes exact rational coefficients in $\mathbb Q[c_1,\dots,c_6]$-    from Formula CITE{formula-component}. It is checked against the displayed terms-    in CITE{ToddWiki} and CITE{GenusWiki}, and by substituting the Chern classes of-    $T\mathbb P^m$ for $1\leq m\leq7$, which gives Todd genus $1$ for each complex-    projective space CITE{GenusWiki}.+  complete-note: it holds every entry with $1\leq n\leq 7$; $\mathrm{Td}_8$ is the+    first component needing more than six variables, which is the most a stored polynomial+    may have+  rigour details: Every stored component was checked against the terms printed in+    CITE{ToddWiki} and CITE{GenusWiki}, and by substituting the Chern classes of $T\mathbb+    P^m$ for $1\leq m\leq7$, which gives Todd genus $1$ CITE{GenusWiki}. The generator+    computes exact rational coefficients from Formula CITE{formula-component}. Display properties:   number-header: $\mathrm{Td}_n$ Numbers:-  '1': 1/2*c1-  '2': 1/12*c1^2 + 1/12*c2-  '3': 1/24*c1*c2-  '4': -1/720*c1^4 + 1/180*c1^2*c2 + 1/240*c2^2 + 1/720*c1*c3 - 1/720*c4-  '5': -1/1440*c1^3*c2 + 1/480*c1*c2^2 + 1/1440*c1^2*c3 - 1/1440*c1*c4-  '6': 1/30240*c1^6 - 1/5040*c1^4*c2 + 11/60480*c1^2*c2^2 + 1/12096*c1^3*c3 + 1/6048*c2^3+- params:+    n: '1'+  number: 1/2*c1+- params:+    n: '2'+  number: 1/12*c1^2 + 1/12*c2+- params:+    n: '3'+  number: 1/24*c1*c2+- params:+    n: '4'+  number: -1/720*c1^4 + 1/180*c1^2*c2 + 1/240*c2^2 + 1/720*c1*c3 - 1/720*c4+- params:+    n: '5'+  number: -1/1440*c1^3*c2 + 1/480*c1*c2^2 + 1/1440*c1^2*c3 - 1/1440*c1*c4+- params:+    n: '6'+  number: 1/30240*c1^6 - 1/5040*c1^4*c2 + 11/60480*c1^2*c2^2 + 1/12096*c1^3*c3 + 1/6048*c2^3     + 11/60480*c1*c2*c3 - 1/12096*c1^2*c4 - 1/60480*c3^2 - 1/6720*c2*c4 - 1/30240*c1*c5     + 1/30240*c6-  '7': 1/60480*c1^5*c2 - 1/12096*c1^3*c2^2 - 1/60480*c1^4*c3 + 1/12096*c1*c2^3 + 11/120960*c1^2*c2*c3-    + 1/60480*c1^3*c4 - 1/120960*c1*c3^2 - 1/13440*c1*c2*c4 - 1/60480*c1^2*c5 + 1/60480*c1*c6+- params:+    n: '7'+  number: 1/60480*c1^5*c2 - 1/12096*c1^3*c2^2 - 1/60480*c1^4*c3 + 1/12096*c1*c2^3+    + 11/120960*c1^2*c2*c3 + 1/60480*c1^3*c4 - 1/120960*c1*c3^2 - 1/13440*c1*c2*c4+    - 1/60480*c1^2*c5 + 1/60480*c1*c6