Stickelberger and the Eigenvalue Theorem IPM MATHEMATICS COLLOQUIUM David A. Cox Amherst College Department of Mathematics & Statistics April 14, 2021 David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 1/18 The Eigenvalue Theorem Let F be algebraically closed and I F[x1,..., xn] be an ideal with n ⊂ V(I) F is finite (so I is zero dimensional). For h F[x ,..., xn], ⊆ ∈ 1 multiplication by h induces a linear map m : F[x ,..., xn]/I F[x ,..., xn]/I. h 1 −→ 1 The Eigenvalue Theorem The eigenvalues of m are h(a) for a V(I). h ∈ I love this theorem because it reveals a wonderful link between polynomial algebra and linear algebra. To actually find solutions, note: The eigenvalues of mxi are the ith coordinates of the solutions. Since mxi mxj = mxi xj = mxj xi = mxj mxi , simultaneous eigenvectors exist. They give the solutions! Recent work of Simon Telen explores how the Eigenvalue Theorem interacts with numerical algebraic geometry. In this lecture, I will instead focus on a name attached to the theorem. David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 2/18 The Eigenvalue Theorem Let F be algebraically closed and I F[x1,..., xn] be an ideal with n ⊂ V(I) F is finite (so I is zero dimensional). For h F[x ,..., xn], ⊆ ∈ 1 multiplication by h induces a linear map m : F[x ,..., xn]/I F[x ,..., xn]/I. h 1 −→ 1 The Eigenvalue Theorem The eigenvalues of m are h(a) for a V(I). h ∈ I love this theorem because it reveals a wonderful link between polynomial algebra and linear algebra. To actually find solutions, note: The eigenvalues of mxi are the ith coordinates of the solutions. Since mxi mxj = mxi xj = mxj xi = mxj mxi , simultaneous eigenvectors exist. They give the solutions! Recent work of Simon Telen explores how the Eigenvalue Theorem interacts with numerical algebraic geometry. In this lecture, I will instead focus on a name attached to the theorem. David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 2/18 The Eigenvalue Theorem Let F be algebraically closed and I F[x1,..., xn] be an ideal with n ⊂ V(I) F is finite (so I is zero dimensional). For h F[x ,..., xn], ⊆ ∈ 1 multiplication by h induces a linear map m : F[x ,..., xn]/I F[x ,..., xn]/I. h 1 −→ 1 The Eigenvalue Theorem The eigenvalues of m are h(a) for a V(I). h ∈ I love this theorem because it reveals a wonderful link between polynomial algebra and linear algebra. To actually find solutions, note: The eigenvalues of mxi are the ith coordinates of the solutions. Since mxi mxj = mxi xj = mxj xi = mxj mxi , simultaneous eigenvectors exist. They give the solutions! Recent work of Simon Telen explores how the Eigenvalue Theorem interacts with numerical algebraic geometry. In this lecture, I will instead focus on a name attached to the theorem. David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 2/18 The Eigenvalue Theorem Let F be algebraically closed and I F[x1,..., xn] be an ideal with n ⊂ V(I) F is finite (so I is zero dimensional). For h F[x ,..., xn], ⊆ ∈ 1 multiplication by h induces a linear map m : F[x ,..., xn]/I F[x ,..., xn]/I. h 1 −→ 1 The Eigenvalue Theorem The eigenvalues of m are h(a) for a V(I). h ∈ I love this theorem because it reveals a wonderful link between polynomial algebra and linear algebra. To actually find solutions, note: The eigenvalues of mxi are the ith coordinates of the solutions. Since mxi mxj = mxi xj = mxj xi = mxj mxi , simultaneous eigenvectors exist. They give the solutions! Recent work of Simon Telen explores how the Eigenvalue Theorem interacts with numerical algebraic geometry. In this lecture, I will instead focus on a name attached to the theorem. David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 2/18 The Eigenvalue Theorem Let F be algebraically closed and I F[x1,..., xn] be an ideal with n ⊂ V(I) F is finite (so I is zero dimensional). For h F[x ,..., xn], ⊆ ∈ 1 multiplication by h induces a linear map m : F[x ,..., xn]/I F[x ,..., xn]/I. h 1 −→ 1 The Eigenvalue Theorem The eigenvalues of m are h(a) for a V(I). h ∈ I love this theorem because it reveals a wonderful link between polynomial algebra and linear algebra. To actually find solutions, note: The eigenvalues of mxi are the ith coordinates of the solutions. Since mxi mxj = mxi xj = mxj xi = mxj mxi , simultaneous eigenvectors exist. They give the solutions! Recent work of Simon Telen explores how the Eigenvalue Theorem interacts with numerical algebraic geometry. In this lecture, I will instead focus on a name attached to the theorem. David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 2/18 Gonzalez-Vega and Trujillo, 1995 in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Theorem 1. (Stickelberger Theorem) Let K F be a field extension with F algebraically closed, h K[x] and ⊂ ∈ J bea zero dimensional ideal in K[x]. If F(J)= ∆ ,...,∆s are the V { 1 } zeros in Fn of J then there exists a basis of F[x]/J such that the matrix of Mh, with respect to this basis, has the following block structure: H1 0 . 0 h(∆i ) ⋆ ... ⋆ 0 H2 . 0 0 h(∆i ) ... ⋆ . where Hi = . . 0 0 . H 0 0 . h(∆ ) s i The dimension of the i-th submatrix is equal to the multiplicity of ∆i as a zero of the ideal J. Notation: M : F[x]/J F[x]/J is multiplication by h. h → David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 3/18 Sottile, 2002 in Computations in Algebraic Geometry with Macaulay2 Stickelberger’s Theorem Let h A and m be as above. Then there is a one-to-one ∈ h correspondence between eigenvectors vξ of mh and roots ξ of I, the eigenvalue of mh on vξ is the value h(ξ) of h at ξ, and the multiplicity of this eigenvalue (on the eigenvector vξ) is the multiplicity of the root ξ. Notation: A = k[X]/I, where I is a zero-dimensional ideal in k[X]. m : A A is multiplication by h A. h → ∈ David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 4/18 Basu, Pollack and Roy, 2006 Algorithms in Real Algebraic Geometry Theorem 4.99 (Stickelberger) For f A, the linear map L has the following properties: ∈ f The trace of Lf is Tr(Lf )= µ(x)f (x) Return ∈ P k x ZerX( ;C ) The determinant of Lf is µ(x) det(Lf )= f (x) Return ∈ P k x ZerY( ;C ) The characteristic polynomial χ( , f , T ) of L is P f χ( , f , T )= (T f (x))µ(x) P − ∈ k x ZerY(P;C ) Notation: A = C[X ,..., X ]/Ideal( , C), C algebraically closed. 1 k P David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 5/18 Goal of This Lecture The last three slides gave versions of a result called “Stickelberger’s Theorem”. The statement of the result was slightly different in each case. One feature they have in common: the authors never cite a specific paper of Stickelberger! Three Questions to Answer What did Stickelberger really do? How does it relate to “Stickelberger’s Theorem”? Why are his papers never cited? David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 6/18 Goal of This Lecture The last three slides gave versions of a result called “Stickelberger’s Theorem”. The statement of the result was slightly different in each case. One feature they have in common: the authors never cite a specific paper of Stickelberger! Three Questions to Answer What did Stickelberger really do? How does it relate to “Stickelberger’s Theorem”? Why are his papers never cited? David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 6/18 Goal of This Lecture The last three slides gave versions of a result called “Stickelberger’s Theorem”. The statement of the result was slightly different in each case. One feature they have in common: the authors never cite a specific paper of Stickelberger! Three Questions to Answer What did Stickelberger really do? How does it relate to “Stickelberger’s Theorem”? Why are his papers never cited? David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 6/18 Goal of This Lecture The last three slides gave versions of a result called “Stickelberger’s Theorem”. The statement of the result was slightly different in each case. One feature they have in common: the authors never cite a specific paper of Stickelberger! Three Questions to Answer What did Stickelberger really do? How does it relate to “Stickelberger’s Theorem”? Why are his papers never cited? David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 6/18 Goal of This Lecture The last three slides gave versions of a result called “Stickelberger’s Theorem”. The statement of the result was slightly different in each case. One feature they have in common: the authors never cite a specific paper of Stickelberger! Three Questions to Answer What did Stickelberger really do? How does it relate to “Stickelberger’s Theorem”? Why are his papers never cited? David A. Cox (Amherst College) Stickelberger & Eigenvalues April14,2021 6/18 Goal of This Lecture The last three slides gave versions of a result called “Stickelberger’s Theorem”. The statement of the result was slightly different in each case. One feature they have in common: the authors never cite a specific paper of Stickelberger! Three Questions to Answer What did Stickelberger really do? How does it relate to “Stickelberger’s Theorem”? Why are his papers never cited? David A.