Efficient Identity Testing and Polynomial Factorization over Non
... Arvind et al. observed that given a monomial m and a homogeneous non-commutative circuit C, one can compute the formal left and right derivatives of C with respect to m efficiently [AJR15]. We need this result too in our algorithm. To sketch the algorithm, consider an easy case when the given polyno ...
... Arvind et al. observed that given a monomial m and a homogeneous non-commutative circuit C, one can compute the formal left and right derivatives of C with respect to m efficiently [AJR15]. We need this result too in our algorithm. To sketch the algorithm, consider an easy case when the given polyno ...
Constructibility of Regular n-Gons
... We now know that a is constructible if and only if there is a tower of fields Q ⊂ K1 ⊂ K2 ⊂ · · · ⊂ Km such that a ∈ Km and the degree of each field extension is 2. Let us formally define degree and relate the degree of the field extension to the nth roots of unity and the polynomial equation xn = 1 ...
... We now know that a is constructible if and only if there is a tower of fields Q ⊂ K1 ⊂ K2 ⊂ · · · ⊂ Km such that a ∈ Km and the degree of each field extension is 2. Let us formally define degree and relate the degree of the field extension to the nth roots of unity and the polynomial equation xn = 1 ...
04 commutative rings I
... Let X be the set of polynomials expressible in the form H − S · M for some polynomial S. Let R = H − Q · M be an element of X of minimal degree. Claim that deg R < deg M . If not, let a be the highest-degree coefficient of R, let b be the highest-degree coefficient of M , and define G = (ab−1 ) · xd ...
... Let X be the set of polynomials expressible in the form H − S · M for some polynomial S. Let R = H − Q · M be an element of X of minimal degree. Claim that deg R < deg M . If not, let a be the highest-degree coefficient of R, let b be the highest-degree coefficient of M , and define G = (ab−1 ) · xd ...