id	sid	tid	token	lemma	pos
n296ww74n34	1	1	in	in	ADP
n296ww74n34	1	2	this	this	DET
n296ww74n34	1	3	dissertation	dissertation	NOUN
n296ww74n34	1	4	,	,	PUNCT
n296ww74n34	1	5	we	we	PRON
n296ww74n34	1	6	will	will	AUX
n296ww74n34	1	7	discuss	discuss	VERB
n296ww74n34	1	8	some	some	DET
n296ww74n34	1	9	properties	property	NOUN
n296ww74n34	1	10	of	of	ADP
n296ww74n34	1	11	pure	pure	ADJ
n296ww74n34	1	12	o	o	NOUN
n296ww74n34	1	13	-	-	NOUN
n296ww74n34	1	14	sequences	sequence	NOUN
n296ww74n34	1	15	,	,	PUNCT
n296ww74n34	1	16	which	which	PRON
n296ww74n34	1	17	,	,	PUNCT
n296ww74n34	1	18	due	due	ADP
n296ww74n34	1	19	to	to	ADP
n296ww74n34	1	20	macaulay	macaulay	NOUN
n296ww74n34	1	21	's	's	PART
n296ww74n34	1	22	inverse	inverse	ADJ
n296ww74n34	1	23	systems	system	NOUN
n296ww74n34	1	24	,	,	PUNCT
n296ww74n34	1	25	are	be	AUX
n296ww74n34	1	26	in	in	ADP
n296ww74n34	1	27	bijective	bijective	ADJ
n296ww74n34	1	28	correspondence	correspondence	NOUN
n296ww74n34	1	29	with	with	ADP
n296ww74n34	1	30	the	the	DET
n296ww74n34	1	31	hilbert	hilbert	PROPN
n296ww74n34	1	32	functions	function	NOUN
n296ww74n34	1	33	of	of	ADP
n296ww74n34	1	34	artinian	artinian	ADJ
n296ww74n34	1	35	level	level	NOUN
n296ww74n34	1	36	monomial	monomial	ADJ
n296ww74n34	1	37	algebras	algebra	NOUN
n296ww74n34	1	38	.	.	PUNCT
n296ww74n34	2	1	in	in	ADP
n296ww74n34	2	2	particular	particular	ADJ
n296ww74n34	2	3	,	,	PUNCT
n296ww74n34	2	4	we	we	PRON
n296ww74n34	2	5	will	will	AUX
n296ww74n34	2	6	focus	focus	VERB
n296ww74n34	2	7	on	on	ADP
n296ww74n34	2	8	their	their	PRON
n296ww74n34	2	9	unimodality	unimodality	NOUN
n296ww74n34	2	10	.	.	PUNCT
n296ww74n34	3	1	a	a	DET
n296ww74n34	3	2	sequence	sequence	NOUN
n296ww74n34	3	3	of	of	ADP
n296ww74n34	3	4	numbers	number	NOUN
n296ww74n34	3	5	is	be	AUX
n296ww74n34	3	6	unimodal	unimodal	ADJ
n296ww74n34	3	7	if	if	SCONJ
n296ww74n34	3	8	it	it	PRON
n296ww74n34	3	9	does	do	AUX
n296ww74n34	3	10	not	not	PART
n296ww74n34	3	11	increase	increase	VERB
n296ww74n34	3	12	after	after	ADP
n296ww74n34	3	13	a	a	DET
n296ww74n34	3	14	strict	strict	ADJ
n296ww74n34	3	15	decrease	decrease	NOUN
n296ww74n34	3	16	.	.	PUNCT
n296ww74n34	4	1	there	there	PRON
n296ww74n34	4	2	has	have	AUX
n296ww74n34	4	3	been	be	AUX
n296ww74n34	4	4	previous	previous	ADJ
n296ww74n34	4	5	progress	progress	NOUN
n296ww74n34	4	6	in	in	ADP
n296ww74n34	4	7	this	this	DET
n296ww74n34	4	8	area	area	NOUN
n296ww74n34	4	9	.	.	PUNCT
n296ww74n34	5	1	specifically	specifically	ADV
n296ww74n34	5	2	,	,	PUNCT
n296ww74n34	5	3	it	it	PRON
n296ww74n34	5	4	is	be	AUX
n296ww74n34	5	5	known	know	VERB
n296ww74n34	5	6	that	that	SCONJ
n296ww74n34	5	7	all	all	DET
n296ww74n34	5	8	algebras	algebra	NOUN
n296ww74n34	5	9	in	in	ADP
n296ww74n34	5	10	two	two	NUM
n296ww74n34	5	11	variables	variable	NOUN
n296ww74n34	5	12	are	be	AUX
n296ww74n34	5	13	unimodal	unimodal	ADJ
n296ww74n34	5	14	,	,	PUNCT
n296ww74n34	5	15	due	due	ADP
n296ww74n34	5	16	to	to	ADP
n296ww74n34	5	17	macaulay	macaulay	NOUN
n296ww74n34	5	18	's	's	PART
n296ww74n34	5	19	maximal	maximal	ADJ
n296ww74n34	5	20	growth	growth	NOUN
n296ww74n34	5	21	theorem	theorem	NOUN
n296ww74n34	5	22	.	.	PUNCT
n296ww74n34	6	1	furthermore	furthermore	ADV
n296ww74n34	6	2	,	,	PUNCT
n296ww74n34	6	3	it	it	PRON
n296ww74n34	6	4	is	be	AUX
n296ww74n34	6	5	known	know	VERB
n296ww74n34	6	6	that	that	SCONJ
n296ww74n34	6	7	all	all	DET
n296ww74n34	6	8	pure	pure	ADJ
n296ww74n34	6	9	o	o	NOUN
n296ww74n34	6	10	-	-	NOUN
n296ww74n34	6	11	sequences	sequence	NOUN
n296ww74n34	6	12	of	of	ADP
n296ww74n34	6	13	type	type	NOUN
n296ww74n34	6	14	two	two	NUM
n296ww74n34	6	15	in	in	ADP
n296ww74n34	6	16	three	three	NUM
n296ww74n34	6	17	variables	variable	NOUN
n296ww74n34	6	18	are	be	AUX
n296ww74n34	6	19	unimodal	unimodal	ADJ
n296ww74n34	6	20	and	and	CCONJ
n296ww74n34	6	21	the	the	DET
n296ww74n34	6	22	hilbert	hilbert	PROPN
n296ww74n34	6	23	function	function	NOUN
n296ww74n34	6	24	of	of	ADP
n296ww74n34	6	25	monomial	monomial	ADJ
n296ww74n34	6	26	complete	complete	ADJ
n296ww74n34	6	27	intersections	intersection	NOUN
n296ww74n34	6	28	are	be	AUX
n296ww74n34	6	29	unimodal	unimodal	ADJ
n296ww74n34	6	30	;	;	PUNCT
n296ww74n34	6	31	these	these	DET
n296ww74n34	6	32	results	result	NOUN
n296ww74n34	6	33	are	be	AUX
n296ww74n34	6	34	due	due	ADJ
n296ww74n34	6	35	to	to	ADP
n296ww74n34	6	36	the	the	DET
n296ww74n34	6	37	weak	weak	ADJ
n296ww74n34	6	38	lefschetz	lefschetz	ADJ
n296ww74n34	6	39	property	property	NOUN
n296ww74n34	6	40	.	.	PUNCT
n296ww74n34	7	1	in	in	ADP
n296ww74n34	7	2	addition	addition	NOUN
n296ww74n34	7	3	to	to	ADP
n296ww74n34	7	4	these	these	DET
n296ww74n34	7	5	families	family	NOUN
n296ww74n34	7	6	of	of	ADP
n296ww74n34	7	7	pure	pure	ADJ
n296ww74n34	7	8	o	o	NOUN
n296ww74n34	7	9	-	-	NOUN
n296ww74n34	7	10	sequences	sequence	NOUN
n296ww74n34	7	11	which	which	PRON
n296ww74n34	7	12	are	be	AUX
n296ww74n34	7	13	unimodal	unimodal	ADJ
n296ww74n34	7	14	,	,	PUNCT
n296ww74n34	7	15	there	there	PRON
n296ww74n34	7	16	are	be	VERB
n296ww74n34	7	17	known	know	VERB
n296ww74n34	7	18	families	family	NOUN
n296ww74n34	7	19	of	of	ADP
n296ww74n34	7	20	pure	pure	ADJ
n296ww74n34	7	21	o	o	NOUN
n296ww74n34	7	22	-	-	NOUN
n296ww74n34	7	23	sequences	sequence	NOUN
n296ww74n34	7	24	which	which	PRON
n296ww74n34	7	25	fail	fail	VERB
n296ww74n34	7	26	to	to	PART
n296ww74n34	7	27	be	be	AUX
n296ww74n34	7	28	unimodal	unimodal	ADJ
n296ww74n34	7	29	.	.	PUNCT
n296ww74n34	8	1	in	in	ADP
n296ww74n34	8	2	particular	particular	ADJ
n296ww74n34	8	3	,	,	PUNCT
n296ww74n34	8	4	for	for	ADP
n296ww74n34	8	5	any	any	DET
n296ww74n34	8	6	r	r	X
n296ww74n34	8	7	>	>	X
n296ww74n34	8	8	2	2	NUM
n296ww74n34	8	9	,	,	PUNCT
n296ww74n34	8	10	there	there	PRON
n296ww74n34	8	11	exists	exist	VERB
n296ww74n34	8	12	a	a	DET
n296ww74n34	8	13	monomial	monomial	ADJ
n296ww74n34	8	14	artinian	artinian	ADJ
n296ww74n34	8	15	level	level	NOUN
n296ww74n34	8	16	algebra	algebra	NOUN
n296ww74n34	8	17	in	in	ADP
n296ww74n34	8	18	r	r	NOUN
n296ww74n34	8	19	variables	variable	NOUN
n296ww74n34	8	20	whose	whose	DET
n296ww74n34	8	21	hilbert	hilbert	NOUN
n296ww74n34	8	22	function	function	NOUN
n296ww74n34	8	23	fails	fail	VERB
n296ww74n34	8	24	unimodality	unimodality	NOUN
n296ww74n34	8	25	with	with	ADP
n296ww74n34	8	26	an	an	DET
n296ww74n34	8	27	arbitrary	arbitrary	ADJ
n296ww74n34	8	28	number	number	NOUN
n296ww74n34	8	29	of	of	ADP
n296ww74n34	8	30	peaks	peak	NOUN
n296ww74n34	8	31	.	.	PUNCT
n296ww74n34	9	1	we	we	PRON
n296ww74n34	9	2	will	will	AUX
n296ww74n34	9	3	focus	focus	VERB
n296ww74n34	9	4	on	on	ADP
n296ww74n34	9	5	the	the	DET
n296ww74n34	9	6	question	question	NOUN
n296ww74n34	9	7	of	of	ADP
n296ww74n34	9	8	whether	whether	SCONJ
n296ww74n34	9	9	particular	particular	ADJ
n296ww74n34	9	10	socle	socle	NOUN
n296ww74n34	9	11	types	type	NOUN
n296ww74n34	9	12	or	or	CCONJ
n296ww74n34	9	13	socle	socle	NOUN
n296ww74n34	9	14	degrees	degree	NOUN
n296ww74n34	9	15	in	in	ADP
n296ww74n34	9	16	a	a	DET
n296ww74n34	9	17	fixed	fix	VERB
n296ww74n34	9	18	number	number	NOUN
n296ww74n34	9	19	of	of	ADP
n296ww74n34	9	20	variables	variable	NOUN
n296ww74n34	9	21	guarantee	guarantee	VERB
n296ww74n34	9	22	unimodality	unimodality	NOUN
n296ww74n34	9	23	.	.	PUNCT
n296ww74n34	10	1	since	since	SCONJ
n296ww74n34	10	2	the	the	DET
n296ww74n34	10	3	weak	weak	ADJ
n296ww74n34	10	4	lefschetz	lefschetz	ADJ
n296ww74n34	10	5	property	property	NOUN
n296ww74n34	10	6	is	be	AUX
n296ww74n34	10	7	only	only	ADV
n296ww74n34	10	8	guaranteed	guarantee	VERB
n296ww74n34	10	9	to	to	PART
n296ww74n34	10	10	hold	hold	VERB
n296ww74n34	10	11	in	in	ADP
n296ww74n34	10	12	the	the	DET
n296ww74n34	10	13	cases	case	NOUN
n296ww74n34	10	14	mentioned	mention	VERB
n296ww74n34	10	15	above	above	ADV
n296ww74n34	10	16	,	,	PUNCT
n296ww74n34	10	17	we	we	PRON
n296ww74n34	10	18	will	will	AUX
n296ww74n34	10	19	use	use	VERB
n296ww74n34	10	20	different	different	ADJ
n296ww74n34	10	21	techniques	technique	NOUN
n296ww74n34	10	22	in	in	ADP
n296ww74n34	10	23	our	our	PRON
n296ww74n34	10	24	approach	approach	NOUN
n296ww74n34	10	25	.	.	PUNCT
n296ww74n34	11	1	we	we	PRON
n296ww74n34	11	2	will	will	AUX
n296ww74n34	11	3	show	show	VERB
n296ww74n34	11	4	that	that	SCONJ
n296ww74n34	11	5	all	all	DET
n296ww74n34	11	6	pure	pure	ADJ
n296ww74n34	11	7	o	o	NOUN
n296ww74n34	11	8	-	-	NOUN
n296ww74n34	11	9	sequences	sequence	NOUN
n296ww74n34	11	10	of	of	ADP
n296ww74n34	11	11	type	type	NOUN
n296ww74n34	11	12	three	three	NUM
n296ww74n34	11	13	in	in	ADP
n296ww74n34	11	14	three	three	NUM
n296ww74n34	11	15	variables	variable	NOUN
n296ww74n34	11	16	and	and	CCONJ
n296ww74n34	11	17	all	all	DET
n296ww74n34	11	18	pure	pure	ADJ
n296ww74n34	11	19	o	o	NOUN
n296ww74n34	11	20	-	-	NOUN
n296ww74n34	11	21	sequence	sequence	NOUN
n296ww74n34	11	22	of	of	ADP
n296ww74n34	11	23	type	type	NOUN
n296ww74n34	11	24	two	two	NUM
n296ww74n34	11	25	in	in	ADP
n296ww74n34	11	26	four	four	NUM
n296ww74n34	11	27	variables	variable	NOUN
n296ww74n34	11	28	are	be	AUX
n296ww74n34	11	29	strictly	strictly	ADV
n296ww74n34	11	30	unimodal	unimodal	ADJ
n296ww74n34	11	31	.	.	PUNCT
n296ww74n34	12	1	we	we	PRON
n296ww74n34	12	2	will	will	AUX
n296ww74n34	12	3	also	also	ADV
n296ww74n34	12	4	show	show	VERB
n296ww74n34	12	5	that	that	SCONJ
n296ww74n34	12	6	all	all	DET
n296ww74n34	12	7	pure	pure	ADJ
n296ww74n34	12	8	o	o	NOUN
n296ww74n34	12	9	-	-	NOUN
n296ww74n34	12	10	sequences	sequence	NOUN
n296ww74n34	12	11	with	with	ADP
n296ww74n34	12	12	socle	socle	NOUN
n296ww74n34	12	13	degree	degree	NOUN
n296ww74n34	12	14	less	less	ADJ
n296ww74n34	12	15	than	than	ADP
n296ww74n34	12	16	or	or	CCONJ
n296ww74n34	12	17	equal	equal	ADJ
n296ww74n34	12	18	to	to	ADP
n296ww74n34	12	19	nine	nine	NUM
n296ww74n34	12	20	in	in	ADP
n296ww74n34	12	21	three	three	NUM
n296ww74n34	12	22	variables	variable	NOUN
n296ww74n34	12	23	are	be	AUX
n296ww74n34	12	24	unimodal	unimodal	ADJ
n296ww74n34	12	25	and	and	CCONJ
n296ww74n34	12	26	all	all	DET
n296ww74n34	12	27	pure	pure	ADJ
n296ww74n34	12	28	o	o	NOUN
n296ww74n34	12	29	-	-	NOUN
n296ww74n34	12	30	sequences	sequence	NOUN
n296ww74n34	12	31	(	(	PUNCT
n296ww74n34	12	32	except	except	SCONJ
n296ww74n34	12	33	for	for	ADP
n296ww74n34	12	34	possibly	possibly	ADV
n296ww74n34	12	35	two	two	NUM
n296ww74n34	12	36	cases	case	NOUN
n296ww74n34	12	37	)	)	PUNCT
n296ww74n34	12	38	with	with	ADP
n296ww74n34	12	39	socle	socle	NOUN
n296ww74n34	12	40	degree	degree	NOUN
n296ww74n34	12	41	less	less	ADJ
n296ww74n34	12	42	than	than	ADP
n296ww74n34	12	43	or	or	CCONJ
n296ww74n34	12	44	equal	equal	ADJ
n296ww74n34	12	45	to	to	ADP
n296ww74n34	12	46	four	four	NUM
n296ww74n34	12	47	in	in	ADP
n296ww74n34	12	48	four	four	NUM
n296ww74n34	12	49	variables	variable	NOUN
n296ww74n34	12	50	are	be	AUX
n296ww74n34	12	51	unimodal	unimodal	ADJ
n296ww74n34	12	52	.	.	PUNCT
n296ww74n34	13	1	finally	finally	ADV
n296ww74n34	13	2	we	we	PRON
n296ww74n34	13	3	will	will	AUX
n296ww74n34	13	4	show	show	VERB
n296ww74n34	13	5	that	that	SCONJ
n296ww74n34	13	6	for	for	ADP
n296ww74n34	13	7	r	r	NOUN
n296ww74n34	13	8	greater	great	ADJ
n296ww74n34	13	9	than	than	ADP
n296ww74n34	13	10	or	or	CCONJ
n296ww74n34	13	11	equal	equal	ADJ
n296ww74n34	13	12	to	to	ADP
n296ww74n34	13	13	4	4	NUM
n296ww74n34	13	14	and	and	CCONJ
n296ww74n34	13	15	e	e	NUM
n296ww74n34	13	16	greater	great	ADJ
n296ww74n34	13	17	than	than	ADP
n296ww74n34	13	18	or	or	CCONJ
n296ww74n34	13	19	equal	equal	ADJ
n296ww74n34	13	20	to	to	ADP
n296ww74n34	13	21	7	7	NUM
n296ww74n34	13	22	,	,	PUNCT
n296ww74n34	13	23	there	there	PRON
n296ww74n34	13	24	exists	exist	VERB
n296ww74n34	13	25	a	a	DET
n296ww74n34	13	26	non	non	ADJ
n296ww74n34	13	27	-	-	ADJ
n296ww74n34	13	28	unimodal	unimodal	ADJ
n296ww74n34	13	29	pure	pure	ADJ
n296ww74n34	13	30	o	o	NOUN
n296ww74n34	13	31	-	-	NOUN
n296ww74n34	13	32	sequence	sequence	NOUN
n296ww74n34	13	33	in	in	ADP
n296ww74n34	13	34	r	r	NOUN
n296ww74n34	13	35	variables	variable	NOUN
n296ww74n34	13	36	with	with	ADP
n296ww74n34	13	37	socle	socle	NOUN
n296ww74n34	13	38	degree	degree	NOUN
n296ww74n34	13	39	e.	e.	PROPN