Quillen-Suslin

JBrett — Mon, 06 Sep 2010 21:11:27 GMT

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	== To Do List for after the Workshop ==		== To Do List for after the Workshop ==

-	*'''~~horrocksTheorem~~ (Branden, Brett)''' - Given a unimodular row f over R = (k[x_1,...x_n]_m)[t], use an inductive process and Suslin's Lemma to find a unimodular matrix U over R such that fU = (1 0 ... 0).	+	*'''horrocks (Branden, Brett)''' - Given a unimodular row f over R = (k[x_1,...x_n]_m)[t], use an inductive process and Suslin's Lemma to find a unimodular matrix U over R such that fU = (1 0 ... 0).
		+	** '''Status: Done.'''
	*'''suslinLemma (Branden, Brett)''' - Given a commutative ring R and polynomials f,g in B[y] with f(y) monic, deg(f) = s, deg(g) <= s-1, and one of the coefficients of g a unit in B, find a polynomial in the ideal (f,g) whose leading coefficient is 1.		*'''suslinLemma (Branden, Brett)''' - Given a commutative ring R and polynomials f,g in B[y] with f(y) monic, deg(f) = s, deg(g) <= s-1, and one of the coefficients of g a unit in B, find a polynomial in the ideal (f,g) whose leading coefficient is 1.
		+	** '''Status: Done.'''
	*'''maximalIdealContaining (Hiro, Branden)''' - Given an ideal I of A = R[x_1,...,x_n] with R a PID, find a maximal ideal of A containing I.		*'''maximalIdealContaining (Hiro, Branden)''' - Given an ideal I of A = R[x_1,...,x_n] with R a PID, find a maximal ideal of A containing I.
	**'''This is partly done thanks to Jason. We still need to implement this using Janet bases or something similar.'''		**'''This is partly done thanks to Jason. We still need to implement this using Janet bases or something similar.'''
	**Description: If k = complex numbers, then, given a set of generators I = (r_1,...,r_k) we can find a solution (a_1,...,a_n) to the system defined by {r_i = 0}. Then using the Nullstellensatz we know that m = (x_1-a_1,...,x_n-a_n) is a maximal ideal containing I. This seems to be more complicated over fields that are not algebraically closed, and even more complicated over Z. See the first section of A. Fabianska's dissertation.		**Description: If k = complex numbers, then, given a set of generators I = (r_1,...,r_k) we can find a solution (a_1,...,a_n) to the system defined by {r_i = 0}. Then using the Nullstellensatz we know that m = (x_1-a_1,...,x_n-a_n) is a maximal ideal containing I. This seems to be more complicated over fields that are not algebraically closed, and even more complicated over Z. See the first section of A. Fabianska's dissertation.
-	*'''~~patchingStep~~ (Brett)''' - Given a set of local solutions to the unimodular row problem, use the patching process described in the Logar-Sturmfels paper to patch them together for a solution over the original polynomial ring.	+	*'''patch (Brett)''' - Given a set of local solutions to the unimodular row problem, use the patching process described in the Logar-Sturmfels paper to patch them together for a solution over the original polynomial ring.
		+	** '''Status: Done.'''
	*'''inductiveStep (Brett)''' - Reduce a unimodular row to a unimodular row involving one less variable.		*'''inductiveStep (Brett)''' - Reduce a unimodular row to a unimodular row involving one less variable.
	*'''qsAlgorithmRow (Brett)''' - Given a unimodular row f over k[x_1,...,x_n], find a square unimodular matrix U such that fU = (1 0 ... 0).		*'''qsAlgorithmRow (Brett)''' - Given a unimodular row f over k[x_1,...,x_n], find a square unimodular matrix U such that fU = (1 0 ... 0).

Macaulay2 - Recent changes [en]

Quillen-Suslin