A Large Polynomial System

The polynomial system is: P9 = {v5, ..., v19} = 0, where

No. of terms Total degree MaxLIC*

v5 13 4 2

v7 49 6 4

v9 131 8 6

v11 292 10 9

v13 577 12 13

v15 1046 14 17

v17 1775 16 22

v19 2859 18 27

P7 = {v5, ..., v15} = 0 if and only if one of the following holds:

• Q1 = 0;
• Q2 = 0, f1 0;
• Q3 = 0, f1 · f2 0;
• Q4 = 0, f1 · f2 · f3 0;
• Q5 = 0, a03 · f1 · f2 · f3 0;
• Q6 = 0, a03 · f1 · f2 · f3 · f4 0.

Q6 consists of six polynomials, of which the following four in a20, a11, a02, a03 are very large:

No. of terms Total degree MaxLIC

w9 458 26 16

w11 539 28 18

w13 1102 36 24

w15 1946 44 32

Open Challenge

• Decompose the polynomial set Q1 and systems [Q2, {f1}], [Q3, {f1, f2}], [Q6, {a03, f1, ..., f4}] into triangular systems.
• Decompose the algebraic variety defined by P7 = 0 into irreducible components, using or without using the Qi.

Please send your comments, suggestions, and solutions to Dongming.Wang@imag.fr.

Known Results

• All the known solutions of P9 = 0 are C1 = 0, K2 = 0 and K4 = {a20, a02, a03} = 0.
• {v5, ..., v19, ...} = 0 has no other real solutions of positive dimension.
• C1 can be decomposed into three irreducible components K1, K3 = {a11, a03} and C0.
• K2 and K1 can be obtained from [Q4, {f1, f2, f3}] and [Q5, {a03, f1, f2, f3}] respectively.
• C0 is contained in Q6.

Conjecture. C1 = 0, K2 = 0, and K4 = 0 cover all the real solutions of {v5, ..., v19, ...} = 0.

Background

• Kukles (1944) showed that v5 = v7 = ... = v19 = ... = 0 "if and only if" one of the following holds: K1 = 0, K2 = 0, K3 = 0, K4 = 0.
• The incompleteness of Kukles' solutions were discovered by Cherkas (1978), Jin and Wang (1990), and Christopher and Lloyd (1990).
• The above results were obtained by Christopher, Lloyd and Pearson (1990-94) and Wang (1996-97).

For further information see JSC 28 (1999): 303-315 or contact Dongming Wang.