Contracting One Specific Variable

The lemmata in this subsection describe situations in which one contracts sets of variables to one variable with specific properties.

Lemma 14. Let K be a set of indices of E^l variables. If ∣K∣ ≥2p−1 and q(K) ≥p, then K contains a contraction J to a variable at level at least l+1, such that J contains variables

of at least two different colours.

Proof. This is a restatement of [14, Lemma 3].

Lemma 15. LetS∈ {C, P}. A set of 2p−1 variables of type S^l contracts to anS^l+1 variable.

Proof. Either one of the S^l variables is already a variable of typeS^l+1 or Lemma 10 can be used withn=2 to show that the set of indices of the 2p−1 variables of type S^l contains a contraction to a variable at level at leastl+1 which can be traced back to at least one of the S^l variables. Therefore, it is anS^l+1 variable.

Lemma 16. LetS∈ {C, P} and let there be3p−2variables of type S^l. Then one can contract them to a variable of type S^l+1, using at most p of them.

Proof. Either one of theS^lvariables is already a variable of typeS^l+1 or, due to Lemma 11, one can contract theS^l variables to a variable at level at leastl+1 using at mostp of them. This variable can be traced back to at least one of theS^l variables, thus it is an S^l+1 variable.

Lemma 17. Let there be 3p−2 variables of type E_ν^l for p≥5 and 2p−1 variables of type E_ν^l for p=5. Then one can contract at most p of these variables to a variable of type E^l+1. Proof. Forp≥5 see [21, Lemma 3.10] and for p=5 see [20, Lemma 3.8].

Lemma 18. Let there be 3p−2 variables of type E_νµ^l forp≥5 or2p−1 variables for p=5.

Then one can contract at most p variables to a variable of type E_ν^l+1.

Proof. LetK be the set of indices of these variables. Letc_i be the corresponding integer of the variablexi. Due to Lemma 12 for p≥5 and Lemma 13 for p=5, there is a non-empty subsetJ⊂K with∣J∣ ≤p, such that∑_j∈Jc_j ≡0 modp while∑_j∈Jc_j≢0 modp² and it follows that

∑

j∈J

( a˜j

˜bj) ≡ ∑

j∈J

c_j(e_ν+µe^ν) ≡ (e_ν+µe^ν) ∑

j∈J

c_j≢0 modp², while ∑_j∈Jc_j≡0 modp. Asp∣e^ν, this leaves

∑

j∈J

(˜a_j

˜b_j) ≡eν∑

j∈Jcj≡pceν modp²

for somecnot congruent to 0 modulo p. Hence, by settingx_i=1 for alli∈J, one can see that J is a contraction of at mostp variables to a variable of typeE^l+1_ν .

Lemma 19. Let there bep−1variables of typeE_νµ^l ₁ and one of typeE_νµ^l ₂ with µ₁≠µ₂. Then one can contract them to an E_ν^l+1_¯ variable.

Proof. Define x⁻¹ for an integer x ∈ Z/pZ as the element in {1, . . . , p−1} which solves x⋅x⁻¹≡1 modp.

Let K be the set of indices of those p variables and ci be the corresponding integer for i∈K. Letx_i0 be theE_νµ^l ₂ variable. Due to Lemma 9 there is a solution of

∑

i∈K

ciy_i^k≡tp modp²

for somet∈ {1, . . . , p}withyi0 ≢0 modp. Consequently, one hasy^k_i0 ≡1 modpbecausep−1∣k

Proof. Either one of theS_ν^l variables is already a S_ν^l+1 variable, or one can assume that they are all of typeE_ν^l as well. The casesl>0 can be reduced to the case l=0 by working with the level coefficient vector(^˜a_˜ⁱ

bi

)instead of the coefficient vector (^ai

bi). See [20, Lemma 3.7] for the casel=0.

Lemma 21. Let H be a set of indices of variables of type E_ν^l with ∣H∣ ≥4p−3 and either for alli∈H the corresponding integer c_i is congruent to an element in the set {1,2, . . . ,^p−1₂ } modulo p or all c_i are congruent to elements in the set {^p+1

2 , . . . , p−1}. Then H contains a contraction K to a variable of typeE_ν^l+1, with∣K∣ ≤2p−2.

Proof. For all i ∈ H, let (ν, µi) be the colour nuance of the variable xi and let di ∈ {1,2, . . . , p−1}and f_i∈ {0,1, . . . , p−1} be such that as c_i=d_i+pf_i.

For the proof one can assume that∣H∣ =4p−3. If this is not the case, one can take a subset ofH to obtain the desired result. The first part proves the weaker claim that H contains a

subsetK containing at most 2p variables such that

for some d≢0 modp. By Lemma 10, the setH contains a non-empty subset J such that

∑

where the last equivalence holds due top∣e^ν and the second and third entry in (2.5.1). The first entry shows that this is congruent to 0 modulop. AsJ is a non-empty subset of H, it follows from the fourth entry that∣J∣ ∈ {p,2p,3p}. If ∣J∣ =3p, take a subset ˜J⊂J containing

which is congruent to 0 modulo p as well. Furthermore, both sets ˆJ and J/Jˆare non-empty, and the smallest of them has at most ^3p₂ ≤2p elements. It follows that in every case there is a

non-empty setK ⊂H containing at most 2p elements, such that

Assume now for such a setK that all corresponding integersci are congruent to elements in the set{1,2, . . . ,^p−1₂ } modulop. It follows thatdi lies in the same set for alli∈K. Hence, it

for some d≢0 modp. This proves the weaker claim if all ci are modulop congruent to an element in the set{1, . . . ,^p−1₂ }. Now let allcibe congruent to elements in the set{^p+1

for somed≢0 modp. Assuming that∣K∣ ≥2p−1, there is, according to Lemma 10 withn=2,

which is congruent to 0 modulop, but not necessarily incongruent to 0 modulop². As

∑ impossible for both subsums to be congruent to 0 modulop². The set for which this sum is incongruent to 0 modulop² is a contraction to a variable of type E_ν^l+1.

Both subsets are non-empty and, hence, as alldiare incongruent to 0 modulop, they contain at least 2 elements. Thus, each one has a most 2p−2 elements, which proves the claim.

Lemma 22. Let S ∈ {C, P} and 0≤m ≤p−1. Let there be p+m variables of type S^l and furtherp−m−1 variables of typeE_ν^l. Then one can contract them to an S^l+1 variable.

Proof. If one of theS^l variables is already an S^l+1 variable, the claim is fulfilled. Thus, one can assume that theS^l variables areE^l variables as well. If there arep variables of the same colourµ, then at least one of them is anS^l variables, because there are at mostp−1 variables which are not. Hence, Lemma 20 shows that one can contract them to anS^l+1 variable.

Else, there are at most p−1 variables of the same colour. Let K be the set of indices of all 2p−1 variables. Then, one has I_max(K) ≤p−1, and thus, q(K) ≥p. By Lemma 14, the setK contains a contraction to to a variable at level at least l+1, using at least two different colours. One can trace that variable back to at least one of theS^l variables, because the variables which are not of typeS^l are all of the same colour, which proves the claim.

Lemma 23. LetS∈ {C, P} and0≤m≤p−1. Let there be p−1variables of type E_ν^l,p−m−1 variables of type E_ν^l_¯ and m+1 variables of type S^l. Then one can contract them to an S^l+1 variable.

Proof. If one of the variable of type S^l is already anS^l+1 variable, the claim is fulfilled, thus one can assume that these variables are of typeE^l as well. Furthermore, one can assume that none of theS^l variables is of typeS_ν^l, because else, Lemma 20 can be use to contract the p−1 variables of typeE_ν^l together with theS_ν^l variable to anS^l+1 variable.

Therefore, one can assume that one hasp−1 variables of typeE_ν^l andpvariables of typeE_ν^l_¯

withx2p−1≢0 modp. The existence of such a solution follows from the proof of Theorem 2 by Olson and Mann [32], but not from the statement of the theorem, from which one can only conclude the existence of a solution, but not that one has one withx2p−1 ≢0 modp. Thus, for the convenience of the reader, the following contains a proof that such a solution exists. In essence the proof uses the same methods as the proof by Olson and Mann, but is tailored for this exact case. applying Lemma 9 again, this time to the system

p−1

∑

i=1αix^k_i +Cy₀^k≡0 modp

provides a solutionyi with p∤y0. It follows that xi=yi for 1≤i≤p−1 is also a solution for

(2.5.3) and, therefore, one has a solution of (2.5.2) given byxi =yi with 1≤i≤2p−1 with p∤x_2p−1. This completes the proof.