Status of Problems

What each rating means is explained on the Mathematical Background page.

Rational Preperiodic

Degree	Problem	ℚ		[K:ℚ] = 2
Degree	Problem	Polynomial	Non-polynomial	Polynomial	Non-polynomial
2	Many points	conjecturalPoonen1998	experimentalVishkautsan2026	experimentalWR1994, Poonen1998, Doyle2014, Doyle2018, Doyle2020	open
	Long cycles	conjecturalPoonen1998	experimentalBCHKW2014	experimentalStoll2008, Hutz2013, Doyle2014, Doyle2018	open
	Long tails	conjecturalPoonen1998	experimentalBCHKW2014	experimentalDoyle2014, Doyle2018	open
3	Many points	experimentalBenedetto2009, DoHy2025	open	open	open
	Long cycles	experimentalBenedetto2009, Hutz2026	open	open	open
	Long tails	experimentalBenedetto2009, Hutz2026	open	open	open
4	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
5	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
6	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
7	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
8	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
9	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
10	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
11	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
12	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
13	Many points	experimentalDoHy2025	open	open	open
	Long cycles	experimentalHutz2026	open	open	open
	Long tails	experimentalHutz2026	open	open	open
14	Many points	open	open	open	open
	Long cycles	open	open	open	open
	Long tails	open	open	open	open
15	Many points	open	open	open	open
	Long cycles	open	open	open	open
	Long tails	open	open	open	open

Postcritically Finite

Degree	ℚ		[K:ℚ] = 2
Degree	Polynomial	Non-polynomial	Polynomial	Non-polynomial
2	provenIngram2012	provenLukas2014	open	open
3	provenIngram2012, AMT2020	open	open	open
4	provenFraser2024	open	open	open
5	open	open	open	open
6	open	open	open	open
7	open	open	open	open
8	open	open	open	open
9	open	open	open	open
10	open	open	open	open
11	open	open	open	open
12	open	open	open	open
13	open	open	open	open
14	open	open	open	open
15	open	open	open	open

Small Height Ratio

Degree	ℚ		[K:ℚ] = 2
Degree	Polynomial	Non-polynomial	Polynomial	Non-polynomial
2	experimentalBCHKW2014, Hutz2026	experimentalBCHKW2014, Hutz2026	open	open
3	experimentalBenedetto2009, Hutz2026	experimentalHutz2026	open	open
4	experimentalHutz2026	experimentalHutz2026	open	open
5	experimentalHutz2026	experimentalHutz2026	open	open
6	experimentalHutz2026	open	open	open
7	experimentalHutz2026	open	open	open
8	experimentalHutz2026	open	open	open
9	experimentalHutz2026	open	open	open
10	experimentalHutz2026	open	open	open
11	experimentalHutz2026	open	open	open
12	experimentalHutz2026	open	open	open
13	open	open	open	open
14	open	open	open	open
15	open	open	open	open

Automorphism Groups

Degree	ℚ		[K:ℚ] = 2
Degree	Polynomial	Non-polynomial	Polynomial	Non-polynomial
2	proven1 groupFN1997	proven2 groupsMilnor1993	open0 groups	open0 groups
3	proven2 groupsFN1997	proven5 groupsGHJSX2021	open0 groups	open0 groups
4	proven2 groupsFN1997	proven5 groupsGHJSX2021	open0 groups	open0 groups
5	proven3 groupsFN1997	open2 groups	open0 groups	open0 groups
6	proven2 groupsFN1997	open1 group	open0 groups	open0 groups
7	proven4 groupsFN1997	open1 group	open0 groups	open0 groups
8	proven2 groupsFN1997	open0 groups	open0 groups	open0 groups
9	proven4 groupsFN1997	open0 groups	open0 groups	open0 groups
10	proven3 groupsFN1997	open0 groups	open0 groups	open0 groups
11	uncomputed0 groups	open1 group	open0 groups	open0 groups
12	uncomputed0 groups	open0 groups	open0 groups	open0 groups
13	uncomputed0 groups	open0 groups	open0 groups	open0 groups
14	uncomputed0 groups	open0 groups	open0 groups	open0 groups
15	uncomputed0 groups	open0 groups	open0 groups	open0 groups

Automorphism Groups: Smallest Degree of Each Group

Dimension 1. For each automorphism group with a map in the database, the smallest degree of a map whose automorphism group is exactly that group, or open if there is none yet. For polynomials over ℚ the automorphism groups are completely classified (FN1997; see Mathematical Background), so that column gives the smallest possible degree, or none if no polynomial has that group. Groups are named as in GAP: D_n is dihedral of order 2n (D₃ is written S₃).