We can find a / b = q1Rr1, where qi is the quotient for that step of the algorithm, and r1 is the remainder from that step of the algorithm. By the rules of remainders and division, b > r1. We now find b / r1 = q2Rr2, so our inequality builds to b > r1 > r2. We ...
Buchberger's algorithm for computing strong standard bases (sBBA). A first criterion takes care of useless GCD-polynomials and goes back to Pan: Lemma 13 Pan (1989) Let f,g∈P such that lc(f)|lc(g) or lc(g)|lc(f). Then gpoly(f,g) reduces to zero w.r.t. {f,g}. Proof...
Proof (a) is a well known conclusion in college algebra that is routinely proved by “working backwards” with the division formulas in the definition of a Euclidean pair given in the Introduction. From a more fancy matrix viewpoint, we can prove (a) as follows. Recall that in the proof...
Chern, S.S.: An elementary proof of the existence of isothermal parameters on a surface. Proc. Am. Math. Soc. 6, 771–782 (1955b) Google Scholar Chern, S.S.: Minimal surfaces in an Euclidean space of N dimensions. In: Differential and Combinatorial Topology, Stewart S. Cairns (Hrsg...
Proof First observe that H−1(0,0)=(0,0). Now suppose that H(α,β)=H(γ,δ). Then if α≠0 and β=0, we have either γ=0 or δ=0, and since ‖α‖2=‖γ‖2−‖δ‖2, we see that δ=0. Since Rn−1[t] is a division ring and γ≠0, there exists precisely...