Noether’s problem Asher Auel Department of Mathematics Yale University Colloquium Dartmouth College May 11, 2018 Symmetric polynomials x + y xy x2 + y 2 x2y + xy 2 x3 + y 3 x4y 4 + 2x5y 2 + 2x2y 5 Symmetric polynomials x + y xy x2 + y 2 = (x + y)2 − 2xy x2y + xy 2 = (x + y)xy x3 + y 3 = (x + y)3 − 3x2y − 3xy 2 x4y 4 + 2x5y 2 + 2x2y 5 = (xy)4 + 2(xy)2(x3 + y 3) Symmetric polynomials x + y xy x2 + y 2 = (x + y)2 − 2xy x2y + xy 2 = (x + y)xy x3 + y 3 = (x + y)3 − 3(x2y + 3xy 2) x4y 4 + 2x5y 2 + 2x2y 5 = (xy)4 + 2(xy)2(x3 + y 3) Symmetric polynomials x + y xy x2 + y 2 = (x + y)2 − 2xy x2y + xy 2 = (x + y)xy x3 + y 3 = (x + y)3 − 3(x + y)xy x4y 4 + 2x5y 2 + 2x2y 5 = (xy)4 + 2(xy)2(x + y)3 − 3(x + y)xy Symmetric polynomials x + y = σ1 xy = σ2 2 2 2 x + y = σ1 − 2σ2 2 2 x y + xy = σ1σ2 3 3 3 x + y = σ1 − 3σ1σ2 4 4 5 2 2 5 4 3 2 3 x y + 2x y + 2x y = σ2 + 2σ1σ2 − 6σ1σ2 Fundamental Theorem Newton 1665, Waring 1770, Gauss 1815 Theorem. Any symmetric polynomial in variables x1;:::; xn can be uniquely expressed as a polynomial in the elementary symmetric polynomials σ1 = x1 + x2 + ··· + xn σ2 = x1x2 + x1x3 + ··· + xn−1xn . X σk = xi1 ··· xik 1≤i1<···<ik ≤n . σn = x1x2 ··· xn Newton–Girard formulas Girard 1629, Newton 1666 k k k pk = x1 + x2 + ··· + xn power sums p1 = σ1 2 p2 = σ1 − 2σ2 3 p3 = σ1 − 3σ1σ2 + 3σ3 4 2 2 p4 = σ1 − 4σ1σ2 + 2σ2 + 4σ1σ3 − 4σ4 0 1 σ1 1 0 ··· 0 B2σ2 σ1 1 ··· 0 C B C B3σ σ σ ··· 0 C pk = det B 3 2 1 C B . .. C @ . 1 A kσk σk−1 σk−2 ··· σ1 0 1 1 0 ··· 0 01 B−2 1 1 ··· 0 0C B C B .. C B 0 −1 1 . 1 0C B C B . .. C @ . 1 1A 0 0 0 · · · −1 1 Example. 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; 123; 199;::: Ln = Ln−1 + Ln−2 Lucas numbers p p 1 + 5 1 − 5 φ = ; φ¯ = roots of x2 − x − 1 2 2 n n Ln = φ +φ¯ = det Vieta’a formula α1; α2; : : : ; αn roots of monic polynomial f (x) n n−1 n−1 n f (x) = x − σ1x + σ2x − · · · + (−1) σn 0 1 1 0 ··· 0 01 B−2 1 1 ··· 0 0C B C B .. C B 0 −1 1 . 1 0C B C B . .. C @ . 1 1A 0 0 0 · · · −1 1 Ln = Ln−1 + Ln−2 Lucas numbers p p 1 + 5 1 − 5 φ = ; φ¯ = roots of x2 − x − 1 2 2 n n Ln = φ +φ¯ = det Vieta’a formula α1; α2; : : : ; αn roots of monic polynomial f (x) n n−1 n−1 n f (x) = x − σ1x + σ2x − · · · + (−1) σn Example. 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; 123; 199;::: 0 1 1 0 ··· 0 01 B−2 1 1 ··· 0 0C B C B .. C B 0 −1 1 . 1 0C B C B . .. C @ . 1 1A 0 0 0 · · · −1 1 Vieta’a formula α1; α2; : : : ; αn roots of monic polynomial f (x) n n−1 n−1 n f (x) = x − σ1x + σ2x − · · · + (−1) σn Example. 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; 123; 199;::: Ln = Ln−1 + Ln−2 Lucas numbers p p 1 + 5 1 − 5 φ = ; φ¯ = roots of x2 − x − 1 2 2 0 1 σ1 1 0 ··· 0 0 B2σ σ 1 ··· 0 0 C B 2 1 C n ¯n B .. C Ln = φ +φ = det B3σ3 σ2 σ1 . 1 0 C B C B . .. C @ . σ1 1 A kσk σk−1 σk−2 ··· σ2 σ1 Vieta’a formula α1; α2; : : : ; αn roots of monic polynomial f (x) n n−1 n−1 n f (x) = x − σ1x + σ2x − · · · + (−1) σn Example. 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; 123; 199;::: Ln = Ln−1 + Ln−2 Lucas numbers p p 1 + 5 1 − 5 φ = ; φ¯ = roots of x2 − x − 1 2 2 0 1 1 0 ··· 0 01 B−2 1 1 ··· 0 0C B C n ¯n B .. C Ln = φ +φ = det B 0 −1 1 . 1 0C B C B . .. C @ . 1 1A 0 0 0 · · · −1 1 Additive symmetric polynomial bases partition λ = (λ1; : : : ; λr ) of n monomial symmetric polynomials X λ1 λr mλ = xτ(1) ··· xτ(r) τ2Sr Schur polynomials X t1 tn sλ = x1 ··· xn T tableau λ Sn Fundamental Theorem. C[x1; x2;:::; xn] = C[σ1; σ2; : : : ; σn] Sn C(x1; x2;:::; xn) = C(σ1; σ2; : : : ; σn) Ring of symmetric polynomials Field of symmetric rational functions Algebra Sn symmetric group of permutations of f1; 2;:::; ng C[x1; x2;:::; xn] ring of polynomials in variables x1; x2;:::; xn Sn acts on C[x1; x2;:::; xn] by permuting the variables 2 2 Example. (1234) · (x1x2 + x3 ) = x2x3 + x4 Algebra Sn symmetric group of permutations of f1; 2;:::; ng C[x1; x2;:::; xn] ring of polynomials in variables x1; x2;:::; xn Sn acts on C[x1; x2;:::; xn] by permuting the variables 2 2 Example. (1234) · (x1x2 + x3 ) = x2x3 + x4 Sn Fundamental Theorem. C[x1; x2;:::; xn] = C[σ1; σ2; : : : ; σn] Sn C(x1; x2;:::; xn) = C(σ1; σ2; : : : ; σn) Ring of symmetric polynomials Field of symmetric rational functions Algebra Fundamental Theorem. Any symmetric polynomial can be uniquely expressed as a polynomial in the elementary symmetric polynomials. uniqueness () σ1; σ2; : : : ; σn algebraically independent Sn () C(x1; x2;:::; xn) = C(σ1; σ2; : : : ; σn) purely transcendental over C 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. 1 2 3 4 5 6 7 8 9 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. 4 7 8 5 3 1 9 2 6 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 75831926 1 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 7 5381926 2 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 7 5 318926 3 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 7 5 3 1 8296 4 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 7 5 3 1 8 2695 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 457318269 6 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 53718269 7 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example. Bubble sort 4 5 3178269 8 4 7 8 5 3 1 9 2 6 even Parity of the number of swaps is an invariant of a permutation An Definition. Ring of alternating polynomials C[x1; x2;:::; xn] An Field of alternating rational functions C(x1; x2;:::; xn) Alternating polynomials An ⊂ Sn subgroups of even permutations Example.