Lesson 4.2 Notes File
... f(x) is the same as solving the related polynomial equation, f(x) = 0. • Zero, solution, root ...
... f(x) is the same as solving the related polynomial equation, f(x) = 0. • Zero, solution, root ...
Lecture 1: Lattice ideals and lattice basis ideals
... We now give another interpretation of toric ideals. A subgroup L of Zn is called a lattice. Recall from basic algebra that L is a free abelian group of rank m ≤ n. The binomial ideal IL ⊂ S generated by the binomials fb with b ∈ L is called the lattice ideal of L. Consider for example, the lattice L ...
... We now give another interpretation of toric ideals. A subgroup L of Zn is called a lattice. Recall from basic algebra that L is a free abelian group of rank m ≤ n. The binomial ideal IL ⊂ S generated by the binomials fb with b ∈ L is called the lattice ideal of L. Consider for example, the lattice L ...
10 Rings
... Example. An n-th root of any a ∈ Z is an algebraic integer. It satisfies p(x) = xn −a = 0. Caution: the roots of p(x) are not necessarily of degree n. For instance p(x) = x4 − 1 = (x2 − 1)(x2 + 1) has roots ±1, ±i, the 4-th roots of unity. Clearly ±1 are of degree 1√and ±i √ are of degree 2. However ...
... Example. An n-th root of any a ∈ Z is an algebraic integer. It satisfies p(x) = xn −a = 0. Caution: the roots of p(x) are not necessarily of degree n. For instance p(x) = x4 − 1 = (x2 − 1)(x2 + 1) has roots ±1, ±i, the 4-th roots of unity. Clearly ±1 are of degree 1√and ±i √ are of degree 2. However ...
