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Ehrhart Positivity Ehrhart Positivity Federico Castillo University of California, Davis Joint work with Fu Liu December 15, 2016 Federico Castillo UC Davis Ehrhart Positivity An integral polytope is a polytope whose vertices are all lattice points. i.e., points with integer coordinates. Definition For any polytope P ⊂ Rd and positive integer m 2 N; the mth dilation of P is mP = fmx : x 2 Pg: We define d i(P; m) = jmP \ Z j to be the number of lattice points in the mP: Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the convex hull of a finite set of points. Federico Castillo UC Davis Ehrhart Positivity Definition For any polytope P ⊂ Rd and positive integer m 2 N; the mth dilation of P is mP = fmx : x 2 Pg: We define d i(P; m) = jmP \ Z j to be the number of lattice points in the mP: Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the convex hull of a finite set of points. An integral polytope is a polytope whose vertices are all lattice points. i.e., points with integer coordinates. Federico Castillo UC Davis Ehrhart Positivity Lattice points of a polytope A (convex) polytope is a bounded solution set of a finite system of linear inequalities, or is the convex hull of a finite set of points. An integral polytope is a polytope whose vertices are all lattice points. i.e., points with integer coordinates. Definition For any polytope P ⊂ Rd and positive integer m 2 N; the mth dilation of P is mP = fmx : x 2 Pg: We define d i(P; m) = jmP \ Z j to be the number of lattice points in the mP: Federico Castillo UC Davis Ehrhart Positivity In this example we can see that i(P; m) = (m + 1)2 Example P 3P Federico Castillo UC Davis Ehrhart Positivity Example P 3P In this example we can see that i(P; m) = (m + 1)2 Federico Castillo UC Davis Ehrhart Positivity Theorem[Ehrhart] Let P be a d-dimensional integral polytope. Then i(P; m) is a polynomial in m of degree d: Theorem of Ehrhart (on integral polytopes) Figure: Eugene Ehrhart. Federico Castillo UC Davis Ehrhart Positivity Theorem of Ehrhart (on integral polytopes) Figure: Eugene Ehrhart. Theorem[Ehrhart] Let P be a d-dimensional integral polytope. Then i(P; m) is a polynomial in m of degree d: Federico Castillo UC Davis Ehrhart Positivity Theorem of Ehrhart (on integral polytopes) Figure: Eugene Ehrhart. Theorem[Ehrhart] Let P be a d-dimensional integral polytope. Then i(P; m) is a polynomial in m of degree d: Federico Castillo UC Davis Ehrhart Positivity We study its coefficients. ... however, there is another popular point of view. The fact that i(P; m) is a polynomial with integer values at integer points suggests other forms of expanding it. An alternative basis We can write: m + d m + d − 1 m i(P; m) = h∗(P) + h∗(P) + ··· + h∗(P) : 0 d 1 d d d The h∗ or δ vector. Therefore, we call i(P; m) the Ehrhart polynomial of P: Federico Castillo UC Davis Ehrhart Positivity ... however, there is another popular point of view. The fact that i(P; m) is a polynomial with integer values at integer points suggests other forms of expanding it. An alternative basis We can write: m + d m + d − 1 m i(P; m) = h∗(P) + h∗(P) + ··· + h∗(P) : 0 d 1 d d d The h∗ or δ vector. Therefore, we call i(P; m) the Ehrhart polynomial of P:We study its coefficients. Federico Castillo UC Davis Ehrhart Positivity The fact that i(P; m) is a polynomial with integer values at integer points suggests other forms of expanding it. An alternative basis We can write: m + d m + d − 1 m i(P; m) = h∗(P) + h∗(P) + ··· + h∗(P) : 0 d 1 d d d The h∗ or δ vector. Therefore, we call i(P; m) the Ehrhart polynomial of P:We study its coefficients. ... however, there is another popular point of view. Federico Castillo UC Davis Ehrhart Positivity The h∗ or δ vector. Therefore, we call i(P; m) the Ehrhart polynomial of P:We study its coefficients. ... however, there is another popular point of view. The fact that i(P; m) is a polynomial with integer values at integer points suggests other forms of expanding it. An alternative basis We can write: m + d m + d − 1 m i(P; m) = h∗(P) + h∗(P) + ··· + h∗(P) : 0 d 1 d d d Federico Castillo UC Davis Ehrhart Positivity Additionally it has an algebraic meaning. More on the the h∗ or δ vector. ∗ ∗ ∗ The vector (h0; h1; ··· ; hd ) has many good properties. Theorem(Stanley) ∗ For any lattice polytope P, hi (P) is nonnegative integer. Federico Castillo UC Davis Ehrhart Positivity More on the the h∗ or δ vector. ∗ ∗ ∗ The vector (h0; h1; ··· ; hd ) has many good properties. Theorem(Stanley) ∗ For any lattice polytope P, hi (P) is nonnegative integer. Additionally it has an algebraic meaning. Federico Castillo UC Davis Ehrhart Positivity What is known? 1 The leading coefficient of i(P; m) is the volume vol(P) of P: 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet. 3 The constant term of i(P; m) is always 1: No simple forms known for other coefficients for general polytopes. Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices (0; 0; 0); (1; 0; 0); (0; 1; 0) and (1; 1; 13), its Ehrhart polynomial is 13 1 i(P; n) = n3 + n2− n + 1: 6 6 Back to coefficients of Ehrhart polynomials Federico Castillo UC Davis Ehrhart Positivity 1 The leading coefficient of i(P; m) is the volume vol(P) of P: 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet. 3 The constant term of i(P; m) is always 1: No simple forms known for other coefficients for general polytopes. Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices (0; 0; 0); (1; 0; 0); (0; 1; 0) and (1; 1; 13), its Ehrhart polynomial is 13 1 i(P; n) = n3 + n2− n + 1: 6 6 Back to coefficients of Ehrhart polynomials What is known? Federico Castillo UC Davis Ehrhart Positivity 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet. 3 The constant term of i(P; m) is always 1: No simple forms known for other coefficients for general polytopes. Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices (0; 0; 0); (1; 0; 0); (0; 1; 0) and (1; 1; 13), its Ehrhart polynomial is 13 1 i(P; n) = n3 + n2− n + 1: 6 6 Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i(P; m) is the volume vol(P) of P: Federico Castillo UC Davis Ehrhart Positivity 3 The constant term of i(P; m) is always 1: No simple forms known for other coefficients for general polytopes. Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices (0; 0; 0); (1; 0; 0); (0; 1; 0) and (1; 1; 13), its Ehrhart polynomial is 13 1 i(P; n) = n3 + n2− n + 1: 6 6 Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i(P; m) is the volume vol(P) of P: 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet. Federico Castillo UC Davis Ehrhart Positivity No simple forms known for other coefficients for general polytopes. Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices (0; 0; 0); (1; 0; 0); (0; 1; 0) and (1; 1; 13), its Ehrhart polynomial is 13 1 i(P; n) = n3 + n2− n + 1: 6 6 Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i(P; m) is the volume vol(P) of P: 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet. 3 The constant term of i(P; m) is always 1: Federico Castillo UC Davis Ehrhart Positivity Warning It is NOT even true that all the coefficients are positive. For example, for the polytope P with vertices (0; 0; 0); (1; 0; 0); (0; 1; 0) and (1; 1; 13), its Ehrhart polynomial is 13 1 i(P; n) = n3 + n2− n + 1: 6 6 Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i(P; m) is the volume vol(P) of P: 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet. 3 The constant term of i(P; m) is always 1: No simple forms known for other coefficients for general polytopes. Federico Castillo UC Davis Ehrhart Positivity Back to coefficients of Ehrhart polynomials What is known? 1 The leading coefficient of i(P; m) is the volume vol(P) of P: 2 The second coefficient equals 1=2 of the sum of the normalized volumes of each facet.
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