On Artin's Conjecture: Pairs of Additive Forms

It is established that for every pair of additive forms $f=\sum_{i=1}^s a_i x_i^k, g=\sum_{i=1}^s b_i x_i^k$ of degree $k$ in $s>2k^2$ variables the equations $f=g=0$ have a non-trivial $p$-adic solution for all odd primes $p$.


Introduction
Let k ≥ 1 be a natural number and a i and b i integer coefficient for 1 ≤ i ≤ s. A special case of Artin's conjecture [1] states that the pair of additive equations have a non-trivial p-adic solution for all primes p provided that s > 2k 2 . Davenport and Lewis [4] started to answer the question whether this statement is true by proving that s > 2k 2 variables are sufficient if k is odd, whereas for even k they only obtained the bound s ≥ 7k 3 . Brüdern and Godinho [2] proved that the expected bound s > 2k 2 holds for even k which are not of the shape for p prime and τ ≥ 1 as well. For each of these excluded shapes they proved for all but one prime that a non-trivial p-adic solution exists if s > 2k 2 . The missing primes are p = 2 in the case k = 3⋅2 τ and p if k = p τ (p − 1). Here, they gave the bounds s ≥ 8 3 k 2 for p = 2 and k = 3 ⋅ 2 τ , s ≥ 8k 2 for p = 2 and k = 2 τ , and s ≥ 4k 2 for p ≥ 3 and k = p τ (p − 1). All in all, the bound s ≥ 8k 2 holds for all p and all k.
There was some further progress for p = 2 and k = 2 τ for τ = 1, τ = 2 and τ ≥ 16. For k = 2 the expected bound s > 8 follows from the general result by Dem'yanov [5] that for two quadratic forms f 1 , f 2 in at least nine variables the equations f 1 = f 2 = 0 have a non-trivial p-adic solution for all primes p. Poehler [14] proved for k = 4, that 49 = 3k 2 + 1 variables suffice and Kränzlein [10] showed for k = 2 τ with τ ≥ 16 that the expected 2k 2 + 1 variables are sufficient.
For p ≥ 3 and k = p τ (p − 1) on the other hand, the bound was further sharpened by Godinho and de Souza Neto [6,7] who proved that s ≥ 2 p p−1 k 2 − 2k suffices for p ∈ {3, 5} and if τ ≥ p−1 2 for p ≥ 7 as well. For k = 6 = 3⋅2, the bound s > 2k 2 was reached by Godinho, Knapp and Rodrigues [8] while later Godinho and Ventura [9] showed that this bound suffices for k = 3 τ ⋅ 2 with τ ≥ 2 as well. Therefore, all pairs of diagonal forms of equal degree k in more than 2k 2 variables have a non-trivial 3-adic solution. The aim of this paper is to prove the following theorem, which shows that this statement does not only hold for p = 3 but for all p ≥ 3, by taking care of the degrees k = p τ (p − 1) for p ≥ 5 and τ ≥ 1.
Theorem. Let p ≥ 5 be a prime, τ ≥ 1 and k = p τ (p − 1). Then for a i , b i ∈ Z with 1 ≤ i ≤ s, the equations have a non-trivial p-adic solution for all s > 2k 2 .
This result is part of the author's PhD thesis submitted to the Georg-August Universität Göttingen on 6 November 2020.
This completes the proof of Artin's conjecture for two diagonal forms of the same degree for all primes p ≠ 2. For p = 2 there are only the questions left whether there is a non-trivial 2-adic solution for k = 3 ⋅ 2 τ for τ ≥ 2 and k = 2 τ for 2 ≤ τ ≤ 15 provided that s > 2k 2 . The argument by Kränzlein [10] can be easily applied for the case k = 3 ⋅ 2 τ as well if τ ≥ 16. Thus, only finitely many k remain for which the bound s > 2k 2 is not reached.
The proof of the theorem follows a pattern by Davenport and Lewis [4] while making use of some improvements by Brüdern and Godinho [2]. Section 2 defines an equivalence relation on the set of all systems (1.1), introduced by Davenport and Lewis [4]. This equivalence relation is defined in a way that solubility of (1.1) in Q s p {0} is preserved, which allows to pick representatives with useful properties from each class and prove the existence of a non-trivial p-adic solution only for them. Due to a version of Hensel's lemma, one can show that a system (1.1) has a non-trivial p-adic solution by proving that the congruences have a solution for which the matrix has rank 2 modulo p. Section 3 recalls the notions of coloured variables, introduced by Brüdern and Godinho [2], and contractions which were established by Davenport and Lewis [4]. Together, they are the foundation of the proof. Coloured variables and a refinement of them provide a way to take care of the rank of the matrix (1.3), while contractions are a means to solve the equations (1.2) recursively by lifting solutions modulo p l to solutions modulo p l+1 . Furthermore, this section continues the path laid down by Davenport and Lewis [4] and Brüdern and Godinho [2], which issues more restrictions on the pairs of equations one has to find a solution for. Section 4 is a collection of combinatorial results which are frequently used, directly and indirectly, in the remaining sections. A description on how the notion of coloured variables is used in combination with contractions to obtain a solution of (1.2) such that the matrix (1.3) has rank 2 is contained in Section 5, whereas Section 6 consists of a collection of lemmata which describe situations in which one can lift some solutions modulo p l to solutions of a higher modulus. The remaining two sections contain the actual proof which is divided into Section 7 for the case k = p (p − 1) and Section 8, where the remaining cases with k = p τ (p − 1) and τ ≥ 2 are handled. This division is due to the different modulus in (1.2). For big τ , one has more variables whose coefficients are not both congruent to 0 modulo p τ +1 , which is balanced in the case τ = 1 by a permutation argument.

p-Normalisation
This section will recall an equivalence relation on the set of systems (1.1) which was introduced by Davenport and Lewis [4] in order to choose representatives with specific characteristics.
Define for any pair of additive forms with rational coefficients a i and b i (1 ≤ i ≤ s) a rational number For integers ν i (1 ≤ i ≤ s) consider the pair f ′ = f (p ν1 x 1 , . . . , p νs x s ) , g ′ = g (p ν1 x 1 , . . . , p νs x s ) (2.2) and for rational numbers λ 1 , λ 2 , µ 1 and µ 2 with λ 1 µ 2 − λ 2 µ 1 ≠ 0 the pair f ′′ = λ 1 f + λ 2 g, g ′′ = µ 1 f + µ 2 g. (2.3) If another pairf ,g with rational coefficients can be obtained by a finite succession of the operations (2.2) and (2.3) on the pair f, g, then they are called p-equivalent. If (x ′ 1 , . . . , x ′ s ) is a non-trivial solution of f ′ = g ′ = 0 then (p ν1 x ′ 1 , . . . , p νs x ′ s ) is a non-trivial solution of f = g = 0, whereas if (x 1 , . . . , x s ) is a non-trivial solution for f = g = 0, then one has a non-trivial solution for f ′ = g ′ = 0 as well, given via (p −ν1 x 1 , . . . , p −νs x s ). Therefore, solubility is preserved under the operation (2.2). The same holds for the operation (2.3). Here, one direction is obvious, and the other holds, because the transformation is invertible. Consequently, the existence of a non-trivial solution for f = g = 0 in Q p implies that there is one for all pairsf ,g which are p-equivalent to f, g. It can also be easily deduced from the definition of ϑ (f, g), that if ϑ (f, g) = 0, the same holds for ϑ (f ′ , g ′ ) and ϑ (f ′′ , g ′′ ) and therefore, for the whole p-equivalent class. Definition 1. A pair f, g given by (2.1) with integers coefficients and ϑ (f, g) ≠ 0 is called pnormalized, if the power of p dividing ϑ (f, g) is as small as possible amongst all pairs of forms (2.1) with integer coefficients in the same p-equivalent class.
As each p-equivalent class contains pairs, for which all coefficients a i , b i are integers, it follows that the existence of a non-trivial solution for all p-normalised pairs induces a non-trivial solution for all pairs of forms with rational coefficients a j , b j and ϑ (f, g) ≠ 0. Using a compactness argument, Davenport and Lewis [4] showed that it induces the existence of a solution for all pairs of forms f, g with ϑ (f, g) = 0 as well. Consequently, it suffices to focus on finding non-trivial p-adic solutions for p-normalised pairs f, g in more than 2k 2 variables. The following lemma gives information about the properties of them.

Lemma 2.
A p-normalised pair of additive forms f, g of degree k in s variables can be written as where f i , g i are forms in m i variables, and these sets of variables are disjoint for i = 0, 1, . . . , k − 1. Moreover, each of the m i variables occurs in at least one of f i , g i with a coefficient not divisible by p. One has Moreover, if q i denotes the minimum number of variables appearing in any form λf i + µg i (λ and µ not both divisible by p) with coefficients not divisible by p, then Proof. See [4,Lemma 9].
At least one integer coefficient a i or b i of a variable x i of a p-normalised pair f, g is non-zero, because else one would have ϑ (f, g) = 0. Consequently, there is a maximal power l of p, which divides both a i and b i . Due to the previous lemma, one can deduce, that 0 ≤ l ≤ k − 1 for all variables x i of a p-normalised pair.

Definition 2.
A variable x i of a pair f, g with integer coefficients is said to be at level l if its coefficients a i and b i are both divisible by p l but not both divisible by p l+1 .
By Lemma 2, a p-normalised pair has exactly m l variables at level l for 0 ≤ l ≤ k − 1. The integersã i ,b i are defined for a variable x i at level l with integer coefficients a i , b i viaã i = p −l a i andb i = p −l b i . These integersã i ,b i are the coefficients of the forms f l , g l as defined in Lemma 2 and the vector ãĩ bi is called the level coefficient vector of a variable x i . One can restrict the question of the existence of a non-trivial p-adic solution to one of congruences. To this end, it is useful to adopt the notation k = p τ δk 0 with δ = gcd (k, p − 1), gcd (p, k 0 ) = 1 and by Davenport and Lewis [4] which is used in the following lemma.
have a solution in the integers for which the matrix has rank 2 modulo p, then the equations (1.1) have a non-trivial p-adic solution.
Such a solution is called a non-singular solution. The remainder of the proof of the theorem will focus on finding non-singular solutions for p-normalised pairs f, g.
The next section will introduce the methods used to find non-singular solutions.

Coloured Variables and Contractions
This section will recall the concept of coloured variables, first used by Brüdern and Godinho [2], and refine it in a way such that it meets the requirements of the special case k = p τ (p − 1). It will also describe the method of contractions which was introduced by Davenport and Lewis [4]. Together, both concepts form the foundation of this proof.
To have more control over the non-singularity of a solution of (2.6), Brüdern and Godinho [2] divided the set of variables at level l into p + 1 sets, depending on their level coefficient vector. For that, they defined the vectors e 0 = 1 0 and e ν = ν 1 for ν ∈ {1, . . . , p}. Viewed as vectors in (Z pZ) 2 the vectors define the sets The parameter I l ν of a pair f, g is the number of variables x i at level l of colour ν.
The parameter q l introduced in Lemma 2 denotes the minimum number of variables appearing with a coefficient not divisible by p in any form λf l + µg l with (λ, µ) ≢ (0, 0) modulo p. This is closely related to the concept of coloured variables. By setting λ ≡ 0 modulo p for ν = 0 or µ ≡ −λν for ν ∈ {1, . . . , p} the variables which appear in λf l + µg l with a coefficient divisible by p are exactly those of colour ν. Consequently, if I l ν ≥ I l µ for all 0 ≤ µ ≤ p it follows that I l ν = m l − q l . Define I l max = m l − q l . This notation can be generalised as follows. Definition 4. For a set K of indices i of variables x i at level l define I ν (K ) as the number of i ∈ K with x i of colour ν, I max (K ) = max 0≤ν≤p I ν (K ) and q (K ) = K − I max (K ).
Note, that if K is the set of all indices of variables at level l, then K = m l , I ν (K ) = I l ν , I max (K ) = I l max and q (K ) = q l . From the definition of a non-singular solution it follows, that whether a solution of (2.6) is non-singular depends exclusively on the variables at level 0. If a solution of (2.6) has variables at level 0 of at least two different colours set to a value which is not congruent to 0 modulo p, the corresponding matrix has rank 2 modulo p making it a non-singular solution. To use variables at different levels one can take sets of variables at one level and combine them in a way that they can be seen as a variable of a higher level. This method was introduced by Davenport and Lewis [4] and applied in combination with the notion of coloured variables by Brüdern and Godinho [2]. Definition 5. Let K be a set of indices j with x j at level l. Let h ∈ N with h > l and suppose that there are integers y j with p ∤ y j such that Then K is called a contraction from level l to level at least h. If either ∑ j∈K a j y k j or ∑ j∈K b j y k j is not congruent to 0 modulo p h+1 , then K is called a contraction from level l to level h.
Recall for variables at level l thatã j = p −l a j andb j = p −l b j . Hence, a set K of variables at level l is a contraction to a variable at level at least l + n if there are y j not divisible by p such that If K is a contraction from level l to some level h, one can set x j = y j X 0 for all j in the contraction K . Through this, one obtains a variable X 0 at level h. One says that the variable X 0 can be traced back to the variables x j with j ∈ K . Assume that there are other variables X i at level h with i ∈ {1, . . . , n}, where each of the variables X i is a variable at level h which either occurred in the pair f, g or is the result of a contraction. If the set of indices {0, 1, . . . , n} of the variables X 0 , X 1 , . . . , X n is a contraction to a variable Y at a level at least h + 1, then one says that the variable Y can be traced back not only to the variables X i for i ∈ {0, 1, . . . , n} but also to all the variables that those variables can be traced back to. For example, Y can be traced back to all x j with j ∈ K . Definition 6. A variable is called a primary variable if it can be traced back to two variables at level 0 of different colours.
If one can contract a primary variable at level at least γ, then by setting this contracted variable 1 and everything else zero, one obtains a non-singular solution of (2.6) and therefore a non-trivial p-adic solution.
In some cases the knowledge of the exact level and colour of a variable that was contracted will give quite an advantage. To gain control about this, the concept of coloured variables is not strong enough because it can only give the information whether a certain set of variables at level l is a contraction to a variable at level at least l + 1, but one does not know the behaviour of the variables modulo p l+2 . Therefore, one cannot use it to extract information about the exact level and colour of the contracted variable. To gain this information, one can divide the set of variables of one colour into smaller sets, which consider the level coefficient vectors ãĩ bi not only modulo p but modulo p 2 .
For that, view the vectors e 0 = 1 0 and e ν = ν 1 as vectors in Z p 2 Z 2 and define the vectors e 0 = 0 p and e ν = p 0 for ν ∈ {1, . . . , p − 1}. This enables one to define sets similar to the sets L ν via L νµ ∶= c (e ν + µe ν ) c ∈ Z p 2 Z * for 0 ≤ ν ≤ p and 0 ≤ µ ≤ p − 1. Here again, a level coefficient vector ãĩ bi lies modulo p 2 in exactly one of the disjoint sets L νµ .

Definition 7.
A variable x i is said to be of colour nuance (ν, µ) if the level coefficient vector ã i ,b i interpreted as a vector in Z p 2 Z 2 lies in L νµ . The parameter I l νµ of a pair f, g is the number of variables x i at level l of colour nuance (ν, µ).
For all variables x i of colour nuance (ν, µ) there is a unique integer c i ∈ {1, 2, . . . , p 2 } pZ for which ãĩ bi ≡ c i (e ν + µe ν ) mod p 2 . The integer c i is said to be the corresponding integer to x i . Lemmata 1 and 3 show that it suffices to find a non-singular solution for all p-normalised pairs in order to prove that for any rational coefficients a j , b j the equations (1.1) have a non-trivial solution in Q p . Due to Lemma 2 one already has some information about the number of variables at certain levels and the distribution of these variables in the different colours of p-normalised forms f, g. One can further exploit that every p-equivalence class contains more than just one p-normalised pair. The next lemma shows further properties that are fulfilled by at least one p-normalised pair in each p-equivalence class for which ϑ (f, g) ≠ 0 holds.

Lemma 4.
Each pair of additive forms (2.1), with rational coefficients and ϑ ≠ 0, is p-equivalent to a p-normalised pair f, g possessing the following properties: (i) g 0 contains exactly q 0 variables with coefficients not divisible by p.
(ii) One of f 1 , g 1 contains exactly q 1 variables with coefficients not divisible by p.
(iii) g 0 has the form It follows from the first property, that I 0 max = I 0 0 = m 0 − q 0 . The second property shows, that either I 1 0 = m 1 − q 1 or I 1 p = m 1 − q 1 and therefore, either the colour 0 or the colour p has the most variables at level 1. Note, that it follows from the third property, that p and thus, that As every p-normalised pair is p-equivalent to a p-normalised pair possessing the properties of the previous lemma, it suffices to prove the existence of a non-singular solutions for p-normalised pairs with these properties.
By using only the variables at level 0 it was proved by Brüdern and Godinho [2,Section 4] that a pair f, g for which q 0 is large has a non-singular solution as displayed in the following.
They said that a colour ν is zero-representing if there is a subset K of variables at level 0 of colour ν for some 0 ≤ ν ≤ p, which is a contraction to a variable at level at least γ. The following Lemma is an immediate result from this definition.

Lemma 5.
If a pair f, g has two colours that are zero-representing, then there exists a non-singular solution of (2.6).
Using a theorem of Olson [12], they then provided a lower bound of the amount of variables at level 0 of colour ν which are required in order to ensure that ν is zero-representing. Using these two lemmata and the theorem of Olson [12] again, they concluded the following statement.
Proof. See [2,Lemma 4.4] Therefore, it suffices to focus on p-normalised forms f, g that fulfil the properties of Lemma 4 and have q 0 ≤ 2p γ − 2.

Combinatorial Results
This section contains a collection of lemmata with combinatorial results on congruences modulo p and p 2 for primes p, which will later be convenient for finding contraction in certain sets.

Lemma 8.
Let n > ggT (k, p − 1) and c 1 , . . . , c n be any integers coprime to p. Then, the congruence Proof. This is the special case G = (Z pZ) n of the theorem of Olson [12].

Strategy
This section contains a general description of the remainder of the proof, for which further notation is introduced. Assume for the remainder of this paper that τ ≥ 1 is an integer, p ≥ 5 a prime and k = p τ (p − 1). This will not be repeated in the following but nonetheless assumed in all following lemmata.

Definition 8.
A p-normalised pair of additive forms f, g as in (2.1) is called a proper p-normalised pair if s ≥ 2k 2 + 1, q 0 ≤ 2p τ +1 − 2 and it satisfies the properties of Lemma 4.
The restrictions on k, p and τ show that γ = τ + 1. Therefore, it follows from Lemmata 1, 4 and 7, that it suffices to prove for every proper p-normalised pair f, g that the equations f = g = 0 have a non-trivial p-adic solution.
The bound s ≥ 2k 2 +1 and Lemma 2 show, that a proper p-normalised pair has the lower bounds . . , k − 1} and Lemma 4 provides furthermore To find a non-trivial p-adic solution for a proper p-normalised pair, it suffices, due to Lemma 3, to show that a non-singular solution exists. Using contractions as described in Section 3, this can be done by showing, that one can construct a primary variable at level τ + 1.
In the following there will be two different strategies to construct a primary variable at level at least τ + 1. For the first, one contracts the variables at level 0 to primary variables at level at least 1. Using contractions recursively, one can obtain primary variables at higher levels, until one eventually reaches at least level τ + 1.
The second strategy will be used if I 0 0 ≥ p τ +1 + p τ − 1. By Lemma 6 with γ = τ + 1, it follows that the colour 0 is zero-representing. In this case it suffices to have a contraction to a variable at level at least τ + 1, which can be traced back to at least one variable at level 0 of a different colour than 0. If such a variable can also be traced back to a variable at level 0 of colour 0, the variable is already primary. Else, there is a contraction to another variable at level at least τ + 1, using only the variables at level 0 of colour 0. Setting both of these variables 1 and everything else zero proves, that there is a non-singular solution of f = g = 0.

Definition 9.
A variable which is either a variable at level 0 of a different colour than 0 or can be traced back to one is called colourful.
the goal is to create a colourful variable at level at least τ + 1. The gain of this second strategy are the variables at level 0 of colour 0. To contract primary variables at level at least 1, one usually uses the variables at level 0. If the goal is only to contract colourful variables at level at least 1, it will suffice to use the q 0 variables at level 0 which are colourful. Then, the variables at level 0 of colour 0 can be used to create variables at a higher level, to help contracting the colourful variables to colourful variables at an even higher level, until one eventually contracts them to a colourful variable at level at least τ + 1. This works, because then, one encounters one of the following two scenarios. Either the colourful variable at level at least τ + 1 can be traced back to a variable at level 0 of colour 0. Then one has used one of those variables, which were created using the variables at level 0 of colour 0, some way along the way, and the colourful variable at level at least τ + 1 is also primary. If on the other hand, the colourful variable at level at least τ + 1 cannot be traced back to a variable at level 0 of colour 0, those helpful variables were not needed, to create a colourful variable at level at least τ + 1. Hence, one can create a colourful variable at level at least τ + 1, without using any of the variables at level 0 of colour 0, which still enables one to create a variable at level at least τ + 1, using only those.
The process of creating a colourful or primary variable at level at least τ + 1 will follow the same pattern. If one has a colourful or primary variable at level at least l, either this variable is already at level at least l + 1, or one tries to find a contraction to a variable at level at least l + 1, which contains the colourful or primary variable and thus ensures, that the resulting variable at level l + 1 is colourful or primary, as well. To find such a contraction, one needs to guarantee, that there are other variables at the same level with certain properties. Thus, one differs between the colourful and primary variables, for which one only needs to know a lower bound of their level, and the remaining variables, which will be useful, to contract colourful or primary variables to colourful and primary variables at a higher level. For them it is important to know the precise level they are at. This will be considered by the following notation.
A primary variable at level at least l of colour nuance (ν, µ) will be denoted by P l νµ , whereas a colourful variable which otherwise has the same properties will be denoted by C l νµ . The notation E l νµ will be used to describe a variable at the exact level l of colour nuance (ν, µ). Note that for S ∈ {C, P } a variable of type S l νµ can either be of type S l+1 νµ or of type E l νµ , but not both. It will be said throughout the proof that a set of variables contracts to a variable with certain properties, if one the following cases occur. Either one of the variables in the set is already a variable with the desired properties, or the set of indices of these variables contains a contraction to a variable with these properties. This will help to minimize the amount of cases in which one has to distinguish between an S l νµ variables being of type S l+1 νµ or E l νµ for S ∈ {C, P }. Sometimes one only wants to establish the level and the colour of one variable. Then, this is denoted by P l ν , C l ν or E l ν . If even the colour is of no importance, such a variable is said to be of type P l , C l or E l . In some cases, one has to denote, that a variable of type E l is not of colour ν, or that a variable of type E l ν is not of colour nuance (ν, µ). This is denoted by E l ν and E l νμ , respectively. It will turn out, that the number of C 1 and P 1 variables one can contract the E 0 variables to is at least partly dependent on the parameter q 0 . Therefore, it will be useful to define a further parameter r = r (f, g) for a pair f, g which restricts the area for q 0 to

Contraction Related Auxiliaries
This section is a compilation of settings in which sets of variables contract to variables at a higher level.
6.1. 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 13. 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. Either one of the S l variables is already a variable of type S l+1 or Lemma 9 can be used with n = 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 least l + 1 which can be traced back to at least one of the S l variables. Therefore, it is an S l+1 variable.
Lemma 15. Let S ∈ {C, P } and let there be 3p − 2 variables 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 the S l variables is already a variable of type S l+1 or one can contract the S l variables to a variable at level at least l + 1 using at most p of them due to Lemma 10. This variable can be traced back to at least one of the S l variables, thus it is an S l+1 variable. Proof. Let K be the set of indices of these variables. Let c i be the corresponding integer of the variable x i . Due to Lemma 11 for p ≥ 5 and Lemma 12 for p = 5, there is a non-empty subset J ⊂ K with J ≤ p, such that ∑ j∈J c j ≡ 0 mod p while ∑ j∈J c j ≢ 0 mod p 2 and it follows that for some c not congruent to 0 modulo p. Hence, by setting x i = 1 for all i ∈ J, one can see that J is a contraction of at most p variables to a variable of type E l+1 ν .
Lemma 18. Let there be p − 1 variables of type E l νµ1 and one of type E l νµ2 with µ 1 ≠ µ 2 . Then one can contract them to an E l+1 ν variable.
Let K be the set of indices of those p variables and c i be the corresponding integer for which is divisible by p because e ν is. For ν = 0 one has because p divides neither c i0 nor µ 2 − µ 1 . It follows that the resulting variable lies at level l + 1 and is of colour and else congruent to Hence, again because p divides neither c i0 nor µ 2 − µ 1 one obtains a variable at level l + 1, which is for t ≡ 0 mod p of colour 0 and for t ≢ 0 mod p of colour For the proof one can assume that H = 4p − 3. If this is not the case, one can take a subset of H to obtain the desired result. The first part proves the weaker claim that H contains a subset K containing at most 2p variables such that for some d ≢ 0 mod p. By Lemma 9, the set H contains a non-empty subset J such that where the last equivalence holds due to p e ν and the second and third entry in (6.1). The first entry shows that this is congruent to 0 modulo p. As J is a non-empty subset of H , it follows from the fourth entry, that J ∈ {p, 2p, 3p}. If J = 3p, take a subsetJ ⊂ J containing 3p − 2 elements. By Lemma 9 with n = 3, there is a subsetĴ ⊆J with as before, which again is congruent to 0 modulo p.
and therefore, which is congruent to 0 modulo p as well. Furthermore, both setsĴ and J Ĵ are non-empty, and the smallest of them has at most 3p 2 ≤ 2p elements. It follows, that in every case there is a non-empty set K ⊂ H containing at most 2p elements, such that Assume now for such a set K that all corresponding integers c i are congruent to elements in the set 1, 2, . . . , p−1 2 modulo p. It follows, that d i lies in the same set for all i ∈ K . Hence, it can be deduced from and, the corresponding integers , again. Using the obtained results, there is a subset K ⊂ H with K ≤ 2p and for some d ≢ 0 mod p and, as ãĩ bi lies in the same set L νµ as −ãi −bi , one further has It follows that for some d ≢ 0 mod p and it further holds that This completes the proof for the weaker claim. Now let K ⊂ H be a subset with K ≤ 2p, for some d ≢ 0 mod p. Assuming that K ≥ 2p − 1, there is, according to Lemma 9 with n = 2, a subsetK ⊂ K with K ≤ 2p − 1 and It follows, that which is congruent to 0 modulo p, but not necessarily incongruent to 0 modulo p 2 . As holds as well, one can deduce, that which is again congruent to 0 modulo p. For at least one of those sets, eitherK or K K , the sum is not congruent to 0 modulo p 2 as the sum over all i ∈ K is not, and therefore, it is impossible for both subsums to be congruent to 0 modulo p 2 . The set for which this sum is incongruent to 0 modulo p 2 is a contraction to a variable of type E l+1 ν . Both subsets are non-empty and hence, as all d i are incongruent to 0 modulo p, they contain at least 2 elements. Thus, each one has a most 2p − 2 elements, which proves the claim. 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 13, the set K 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 the S l variables, because the variables which are not of type S l are all of the same colour, which proves the claim. Proof. If one of the variable of type S l is already an S l+1 variable, the claim is fulfilled, thus one can assume that these variables are of type E l as well. Furthermore, one can assume, that none of the S l variables is of type S l ν , because else, Lemma 19 can be use to contract the p − 1 variables of type E l ν together with the S l ν variable to an S l+1 variable. Therefore, one can assume that one has p−1 variables of type E l ν and p variables of type E l ν from which at least one is an S l variable. For convenience name the E l ν variables x 1 , . . . , x p−1 and the E l ν variables x p , . . . , x 2p−1 , where x 2p−1 is an S l variable. Furthermore, let c i be the corresponding integer of x i for 1 ≤ i ≤ 2p − 1 and ν i ≠ ν the colour of the variables x i for p ≤ i ≤ 2p − 1. These 2p − 1 variables contract to an S l+1 variable if there is a solution of The existence of such a solution follows from the proof of Theorem 2 by Olson and Mann [11], 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 with x 2p−1 ≢ 0 mod p. 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. By applying the linear transformation induced by 1 0 1 −ν if ν ≠ 0, one can transform the case ν ≠ 0 to the case ν = 0, because 1 0 1 −ν e ν = νe 0 and 1 0 1 −ν e νi ∈ Lν for someν ≠ ν. All that remains is to solve a system of the kind with y 2p−1 ≢ 0 mod p. This reduces the system (6.2) by setting i . Now consider an additional variable y 0 . If p ∤ y 0 then y k 0 ≡ 1 mod p, hence, applying Lemma 8 again, this time to the system provides a solution y i with p ∤ y 0 . It follows that x i = y i for 1 ≤ i ≤ p − 1 is also a solution for (6.3), and therefore, one has a solution of (6.2) given by This completes the proof. Proof. This is the special case δ = gcd (k, p − 1) = p − 1 of a result from Lemmata 1 and 3 of [4] which is proved in the second paragraph of Section 6 of that paper.
Lemma 24. Let S ∈ {C, P } and let there be x variables of type S l . They contract to ⌈ x+3 p ⌉ − 3 variables of type S l+1 , where each contraction contains at most p variables, leaving at least min{2p− 2, x} variables of type S l unused.
Proof. For x ≤ 3p − 3 the statement is trivial. Therefore, let x ≥ 3p − 2. Assume first, that all x variables are also of type E l . Then there is a contraction of at most p variables to an S l+1 variable due to Lemma 15. Hence, after doing this ⌈ x+3 p ⌉ − 4 times, there are still at least unused S l variables. Hence, one can apply Lemma 15 once more, to obtain ⌈ x+3 p ⌉ − 3 contractions, leaving at least 2p − 2 variables unused. Thus, in this case, the claim holds. Now assume that of the x variables of type S l there are y variables already of type S l+1 while the remaining x − y variables are of type E l . One has because of x ≥ 3p − 2. If x − y ≤ 2p − 2, one can divide the y variables of type S l+1 in one set containing ⌈ x+3 p ⌉ − 3 and one set containing 2p − 2 − (x − y) of them. The variables in the second set together with the remaining x − y variables of type S l are at least 2p − 2 variables of type S l , while the first set contains the ⌈ x+3 p ⌉ − 3 variables of type S l+1 . Thus one can assume, that x − y ≥ 2p − 1 and use the first part of this proof. The set of the x − y variables of type E l contains at least contractions to variables of type S l+1 , leaving at least 2p − 2 variables of type S l unused. Together with the y variables of type S l+1 this gives at least Lemma 25. Let there be x variables of type E l ν . They contract to ⌈ x 2p−2 ⌉ − 4 variables of type E l+1 ν , leaving at least min{6p − 9, x} variables of type E l ν unused. Proof. For x < 8p − 7 the statement is trivial. If x ≥ 8p − 7, one can divide the x variables in two sets. Those for which the corresponding integer c i is congruent to one element in {1, . . . , p−1 2 } modulo p, and the remaining variables. As long as there are at least 8p − 7 variables left, at least one of these sets contains at least 4p − 3 variables, which indicates that one can contract at most 2p − 2 of them to a variable of type E l+1 ν due to Lemma 20. Doing this ⌈ x 2p−2 ⌉ − 5 times leaves at least unused variables, hence, there is another contraction, leaving at least 6p − 9 variables unused.

Lemma 26. A set of
In both cases, this leaves at least 6p − 9 of the E l ν variables unused. Proof. A set of at least (3p − 3) p + 1 variables of type E l ν contains at least 3p − 2 variables which are of the same colour nuance. By Lemma 17, one can contract at most p variables of them to a variable of type E l+1 ν . Repeating this as often as possible provides ⌈ x p ⌉ − 3p + 3 variables of type E l+1 ν and leaves at least unused E l ν variables. For p = 5 this can be done as long as there are at least (2p − 2) p + 1 variables left. Therefore, one can do it ⌈ x p ⌉ − 2p + 2 times, leaving at least unused variables. Using Lemma 25 provides another p + p−1 2 − 4 variables of type E l+1 ν for p ≥ 5 and one for p = 5, while leaving at least 6p − 9 unused variables. All in all, one obtains ⌈ Lemma 27. Let S ∈ {C, P } and x, y and z be non-negative integers with y + z ≥ (2 − m) p − 2 for some m ∈ {0, 1, 2} and x − m ≥ 0. Let there be (p − 1) y variables of type E l ν , (p − 1) y variables of type E l ν and px + y + z variables of type S l . Then one can contract them to x + y − m variables of type S l+1 without using z + mp of the variables of type S l .
Proof. Using Lemma 15 to contract p of the variables of type S l to an S l+1 variable can be done x − m times. This leaves y + z + mp ≥ 2p − 2 variables of type S l . Then, one can construct y sets, each consisting of one S l variable, p − 1 variables of type E l ν and p − 1 variables of type E l ν . By Lemma 22, each of this sets contains a contraction to an S l+1 variable, giving a total of x + y − m variables of type S l+1 as claimed, without using z + mp variables of type S l .
Lemma 28. Let S ∈ {C, P } and x be a non-negative integer. Let K be a set of E l variables with K ≥ (2p − 2) x + p 2 − 3p + 1 and q (K ) ≥ (p − 1) x and let there be further x variables of type S l . Then one can contract them to x variables of type S l+1 .
Proof. The first part of the proof will show via induction on x that the set K contains x distinct sets S i with S i = 2p − 2 and q (S i ) = p − 1 for all 1 ≤ i ≤ x.
For x = 0 the statement is true. It suffices to show for x ≥ 1 that K contains a set H with H = 2p − 2 and q (H ) = p − 1 such that K H ≥ (x − 1) (2p − 2) + p 2 − 3p + 1 and q (K H ) ≥ (x − 1) (p − 1). If such a set H exists, the induction hypothesis ensures that one can find further Thus, one can take H as a set containing p − 1 variables of type E l ν and p − 1 variables of type E l ν from which it follows that H = 2p − 2, q (H ) = p − 1 and For β ≥ p − 1 one has the trivial bound whereas for β ≤ p − 2 it follows that and thus It follows, that the set K contains x distinct sets S i with S i = 2p − 2 and q (S i ) = p − 1.
For each set S i there is a ν i such that I max (S i ) = I νi (S i ) = p − 1. For i ∈ {1, . . . , x} take the set S i and one variable of type S l , which gives p − 1 variables of type E l νi , p − 1 variables of type E lν i and one S l variable. Such a set contains a contraction to an S l+1 variable due to Lemma 22. Thus, one obtains x variables of type S l+1 .
Lemma 29. Let S ∈ {C, P } and x, y and z be non-negative integers with y + z ≥ (2 − m) p − 2 for some m ∈ {0, 1, 2} and x − m ≥ 0. Let there be (2p − 2) y + p 2 − 3p + 1 variables of type E l from which at least (p − 1) y variables are of type E l ν for any 0 ≤ ν ≤ p. Furthermore, let there be px + y + z variables of type S l . Then one can contract them to x + y − m variables of type S l+1 without using z + mp of the variables of type S l .
Proof. Using Lemma 15 to contract p of the variables of type S l to an S l+1 variable can be done x − m times. This leaves y + z + mp ≥ 2p − 2 variables of type S l . One can contract y of them together with the variables of type E l to y variables of type S l+1 due to Lemma 28. This gives a total of x + y − m variables of type S l+1 as claimed, without using z + mp variables of type S l .
Lemma 30. Let x be a non-negative integer. Let there be at least px + p 2 − 3p + 3 variables of type E l ν from which at least x are of type E l νµ for some µ and at least x are of type E l νμ . Then one can contract px of them to x variables of type E l+1 ν .
Proof. Divide the E l ν variables in three sets. One contains x variables of type E l νµ , the next one contains x variables of type E l νμ and the last one contains the remaining variables. The statement is trivial for x = 0, thus one can assume that x ≥ 1. Assume now, that the last set contains z ≥ (p − 2) p + 1 = p 2 − 2p + 1 variables, and the first two both contain y ≥ 1 variables. Then there is an η such that the last set contains at least p − 1 variables of type E l νη and one can choose one variable in one of the first two sets, which is of type E l νη . These p variables contract to an E l+1 ν variable due to Lemma 18. Then, one can take one variable in the untouched set and put it in the last set, such that the first two sets both contain y − 1 variables and the last one contains z − p + 2 variables.
Starting with z ≥ (p − 2) x + p 2 − 3p + 3 and y = x, after following this process x − 1 times, one still has at least p 2 − 2p + 1 variables in the last set left, while the other two each contain one variable. It follows, that one can contract one more variable of type E l+1 ν as described above, giving a total of x variables of type E l+1 ν .
6.3. Inductive Contractions. This subsection uses induction to contract sets of variables at some level to variables more than one level higher. Proof. For i = j the statement is trivial, thus, the cases i < j ≤ τ remain. Assume for an l ∈ {i, . . . , j − 1} that there are p τ −l+1 + mp τ −l − 2 variables of type S l and 2p − 2 variables of type S n for all n ∈ {i, . . . , l − 1}. Lemma 24 shows that these variables can be contracted to variables of type S l+1 . This leaves at least 2p − 2 variables of type S l unused. The claim follows via induction.
Lemma 32. Let S ∈ {C, P } and i, j ∈ N 0 with i ≤ j ≤ τ as well as m ∈ Z with m ≥ −1. Let there be p τ −i+1 + mp τ −i variables of type S i and for all l ∈ {i, . . . , j − 1} let there be an ν l and 2p − 2 variables of type E l ν l . Then one can contract them to p τ −j+1 + mp τ −j variables of type S j . Proof. For i = j the statement is trivial, thus, the cases i < j ≤ τ remain. Assume for an l ∈ {i, . . . , j − 1} there are p τ −l+1 + mp τ −l variables of type S l and 2p − 2 variables of type E l ν l . Lemma 24 shows that there exist contractions to variables S l+1 , each of them containing at most p variables. Therefore, there are even 2p variables of type S l remaining. Together with the 2p − 2 variables of type E l ν l , they can be contracted to another two S l+1 variables, using Lemma 21 twice. This gives a total of p τ −l +mp τ −l−1 variables of type S l+1 . The claim follows via induction.
variables of type E j ν . Then one can contract them to p − m − 1 variables of type E τ ν and 2p − 2 variables of type E i ν for all i ∈ {j, j + 1, . . . , τ − 1}. Proof. If j ≤ τ − 2, assume that for some l ∈ {j, j + 1. . . . , τ − 2} one can contract the variables to variables of type E l+1 ν , while leaving at least 6p − 9 ≥ 2p − 2 variables of type E l ν unused. Hence, by induction, one can contract the E j ν variables to This reduced the cases j ≤ τ − 2 to the case j = τ − 1. For j = τ − 1, one can contract the variables of type E τ −1 variables of type E τ ν with Lemma 25, while leaving at least 6p − 9 ≥ 2p − 2 variables of type E τ −1 ν . This proves the claim.
For j ≤ τ − 2, assume that for some l ∈ {j, j + 1. . . . , τ − 2} one can contract the variables to p i − 2p + 2 variables of type E l ν and 2p − 2 variables of type E i ν for all i ∈ {j, j + 1, . . . , l − 1}. Using Lemma 26 for p = 5, the variables of type E l ν can be contracted to variables of type E l+1 ν , while leaving at least 6p − 9 ≥ 2p − 2 variables of type E l ν unused. By induction, it follows that one can contract variables of type E τ −1 ν and 2p − 2 variables of type E i ν for all i ∈ {j, j + 1, . . . , τ − 2}. This reduced the cases j ≤ τ − 2 to the case j = τ − 1.
For j = τ − 1 one has 3p 2 − mp − 5p + 1 variables of type E j ν . This is at least as big as 2p 2 − 2p + 1 for m ≤ 2. Thus, one can use Lemma 26 for p = 5 to contract them to Proof. For j = τ − 1 the statement is trivial, thus, the cases j ∈ {0, 1, . . . , τ − 2} remain. Assume that for some l ∈ {j, . . . , τ − 2} one can contract the variables of type Then they can be contracted with Lemma 26 to variables of type E l+1 ν , while leaving at least 6p − 9 ≥ 2p − 2 variables of type E l ν . Via induction one can deduce that on can contract 2p − 2 variables of type E i ν for all i ∈ {j, . . . , τ − 2} and

Pairs of Forms with τ = 1
This section contains the proof that for all proper p-normalised pairs f, g with τ = 1 the equations f = g = 0 have a non-trivial p-adic solution. This is primarily done by contracting a C τ +1 = C 2 variable if I 0 0 ≥ p 2 + p − 1, which indicates that the colour 0 is zero-representing, and else by contracting a P τ +1 = P 2 variable.
The following lemma will exploit p-equivalence classes by transforming some pairs f, g into p-equivalent pairsf ,g, for which one can contract a P 2 variable.
Lemma 36. Let 1 ≤ m ≤ p be a natural number and j ∈ {0, . . . , k − 1}. Let f, g be a pair given by Then there exists a non-trivial p-adic solution of f = g = 0.
Proof. Apply x ↦ px for all variables at level l for all l ∈ {0, . . . , j − 1}, and then multiply both equations with p −j . This transforms the pair f, g into a p-equivalent pair with integer coefficients, q 0 ≥ pm, m 0 ≥ m (2p − 1), q 1 ≥ p − m and I 1 ν = I 1 max ≥ p − 1 for some ν. Using Lemma 23, one can contract the E 0 variables to m variables of type P 1 . The p − 1 variables of type E 1 ν and the p − m variables of type E 1 ν can be contracted together with the P 1 variables to a P 2 variable due to Lemma 22. Thus, the transformed pair has a non-trivial p-adic solution, from which it follows that the p-equivalent pair f, g has one as well.
Proof. As describe above, one can assume that q 1 = 0 and thus I 1 Assume first, that r (f, g) = r ≥ 0. Then one can use Lemma 23 to contract the E 0 variables to p variables of type P 1 and Lemma 21 to contract the P 1 variables together with the E 1 ν variables to a P 2 variable. Consequently, one can assume, that r = −1 which leads to Hence, the colour 0 is zero-representing and it suffices to show that one can contract a C 2 variable.
In both cases, there are still at least 2p 2 − 3p + 2 − (2p − 2) = 2p 2 − 5p + 4 ≥ 2p − 2 variables of type E 0 0 remaining. Those contract with p 2 − p of the C 0 variables to p − 1 variables of type C 1 due to Lemma 32. All in all, one has p − 1 variables of type E 1 ν , one E 1 ν variable and p − 1 variables of type C 1 . Due to Lemma 22 these can be contracted to a C 2 variable, which completes the proof.
Lemma 38. Let f, g be a proper p-normalised pair with τ = 1 and I 1 Then the equations f = g = 0 have a non-trivial p-adic solution.
Proof. By I 1 µ ≤ I 1 max ≤ p − 2 for all 0 ≤ µ ≤ p, it follows that If one has q 1 ≥ p and m 1 ≥ 2p − 1 one can assume, due to Lemma 36, that either q 2 ≤ p − 2 or Else, one has either q 1 ≤ p − 1 or m 1 ≤ 2p − 2. If q 1 ≤ p − 1 it follows that m 1 ≤ 2p − 2 as well, because m 1 = I 1 max + q 1 . Then one obtains One of these three bounds holds in any case, thus, one can assume that This lower bound for m 0 leads to For r ≤ p − 2 this is at least as big as p 2 + p − 1 for p ≥ 5, hence, it suffices to contract a C 2 variable, whereas one has to contract a P 2 variable for r = p − 1. The remaining proof will be divided into three cases, based on the value of r = r (f, g). Case r = p − 1. If m 0 ≥ (2p − 1) (2p − 1) = 4p 2 − 4p + 1, one can use Lemma 23 to contract the E 0 variables to 2p − 1 variables of type P 1 . By Lemma 14, it follows that one can contract those P 1 variables to a P 2 variable. Hence, one can assume that m 0 ≤ 4p 2 −4p and thus m 1 ≥ 1. Due to (7.2) one has m 0 ≥ 4p 2 − 6p + 2 = (2p − 1) (2p − 2). Therefore, Lemma 23 shows, that one can contract the E 0 variables to 2p − 2 variables of type P 1 . Lemma 9 with n = 2 shows that one can contract them together with one of the E 1 variables to a variable of a level at least 2. This contraction cannot contain only the E 1 variable, thus the resulting variable has to be a P 2 variable. Case 0 ≤ r ≤ p − 2. One can assume that I 1 ν = I 1 max ≤ p − r − 2, because else, Lemma 32 can be used to contract p 2 + rp of the C 0 variables together with 2p − 2 variables of type E 0 0 to p + r variables of type C 1 . Then one can contract them together with the E 1 ν variables to a C 2 variable due to Lemma 21. It follows that that If q 2 ≥ p − 1 and I 2 max ≥ p − 1, one can use Lemma 16 to contract p (p − r − 1) of the variables of type E 0 0 to p − r − 1 variables of type E 1 . This is possible, because afterwards, there are still at least of the E 0 0 variables unused. Lemma 32 can be used to contract p 2 + rp of the C 0 and 2p − 2 of the remaining E 0 0 variables to p + r variables of type C 1 . One can assume, that the C 1 variables are of type E 1 , because else one already has a C 2 variable. Take the set K of the 2p − 1 variables of type E 1 that were contracted, from which p + r are of type C 1 . If there is a µ with I µ (K ) ≥ p there are at least p variables of type E 1 µ in K . As p + r of the variables in K are type C 1 , it follows that there is at least one C 1 µ variable in K . Thus, one can contract the E 1 µ variables in K with Lemma 19 to a C 2 variable. Else, one has q (K ) ≥ p and thus, one has transformed the pair f, g into another one with m 1 ≥ 2p − 1 and q 1 ≥ p. This new pair has the same values for q 2 and I 2 max , thus it follows from Lemma 36 that it has a non-trivial p-adic solution. Consequently the pair f, g has one as well. Thus, one can assume, that either q 2 ≤ p − 2 or I 2 max ≤ p − 2. By (7.4), it follows for q 2 ≤ p − 2 that m 0 ≥ 5p 2 − 5p + 1 − p 2 + (r + 1) p + r + 2 − p + 2 = 4p 2 − 5p + rp + 5 + r and for I 2 max ≤ p − 2 that m 2 ≤ p 2 − p − 2 and therefore m 0 ≥ 6p 2 − 6p + 1 − p 2 + (r + 1) p + r + 2 − p 2 + p + 2 = 4p 2 − 4p + rp + 5 + r.
In both cases, one obtains the lower bound m 0 ≥ 4p 2 − 5p + rp + 5 + r, which leads to Now one can distinguish between the cases m 1 ≥ 1 and m 1 = 0. Case m 1 ≥ 1. One can use Lemma 25 to contract the E 0 0 variables to variables of type E 1 0 . This leaves at least 6p − 9 ≥ 2p − 2 variables of type E 0 0 . Hence, one can use Lemma 32 to contract them with p 2 + rp of the C 0 variables to p + r variables of type C 1 . The set H , containing the p − r − 2 variables of type E 1 0 , the p + r variables of type C 1 and one further E 1 variables, which exists due to m 1 ≥ 1, contains a contraction to a C 2 variable. If none of the C 1 variables is already of type C 2 , there is either a µ such that I µ (H ) ≥ p or q (H ) ≥ p. If I µ (H ) ≥ p, then at least one of the E 1 µ variables in H is a C 1 variable and thus H contains a contraction to a C 2 variable due to Lemma 19. If on the other hand q (H ) ≥ p, then H contains a contraction to a variable at level at least 2, which can be traced back to at least two variables of different colour at level 1, due to Lemma 13. The only way that such a variable is not of type C 2 , is that the contraction contains no C 1 variable. The variables in H which are not of type C 1 are p − r − 2 variables of type E 1 0 and one E 1 variable. As the contracted variable can be traced back to two variables of different colours at level 1, the E 1 variable has to be an E 1 0 variable. But if a subset K of H contains this variable and additionally only variables of type E 1 0 , then it cannot be a contraction to a variable at level at least 2, because then one has exactly one i ∈ K for which the second entryb i of the level coefficient vector is not congruent to 0 modulo p. Therefore, one cannot solve ∑ j∈Kbj y k j ≡ 0 mod p with all y j ≢ 0 mod p. Consequently, this cannot occur, and the resulting variable is a C 2 variable. Case m 1 = 0. This leads to the even better bound m 0 ≥ 4p 2 − 4p + 1 and thus For p ≥ 7, this is at least as big as 2p 2 + 2p − 2rp + 2r − 3, thus, one can use Lemma 33 to contract the E 0 0 variables to p − r − 1 variables of type E 1 0 , while leaving at least 2p − 2 variables of type E 0 0 unused. For p = 5, this is at least as big as 3p 2 − rp − 5p + 1, thus Lemma 34 shows that one can contract the E 0 0 variables to p − r − 1 variables of type E 1 0 as well, while leaving at least 2p − 2 variables of type E 0 0 unused. In both cases, one can use Lemma 32 to contract the 2p − 2 variables of type E 0 0 with p 2 + rp of the C 0 variables to p + r variables of type C 1 . Then one can contract them together with the p − r − 1 variables of type E 1 0 to a C 2 variable due to Lemma 21. Case r = −1. Note first, that one has m 1 − I 1 0 ≤ p 2 − 2p = (p − 2) p due to I 1 max ≤ p − 2, and thus Therefore, one can take p − 1 variables of type E 0 00 and one of type E 0 00 , which can be contracted to a E 1 0 variable by Lemma 18. There are at least 3p 2 − 5p + 1 variables of type E 0 0 remaining, which can be contracted to p − 1 variables of type E 1 0 using Lemma 25 for p ≥ 7 and Lemma 26 for p = 5. This leaves at least 6p − 9 ≥ 2p − 2 variables of type E 0 0 , which can be contracted with p 2 − p of the C 0 variables to p − 1 variables of type C 1 using Lemma 32. Then one can use Lemma 22 to contract the p − 1 variables of type E 1 0 , the p − 1 variables of type C 1 and the E 1 0 variable to a C 2 variable. Hence, one can assume that It follows that m 1 ≥ 2. Note, that one has Case m 1 − I 1 0 = 0. Due to m 1 ≥ 2, one has I 1 0 ≥ 2. Take a set, which contains p − 1 variables of type E 0 00 and one E 0 00 variable. This set contains a contraction to an E 1 0 variable due to Lemma 18. Then there are at least 3p 2 − 7p + 4 ≥ 2p 2 − 2p + 1 variables of type E 0 0 left. Therefore, one can use Lemma 25 to contract them to p − 3 variables of type E 1 0 , giving a total of p − 1, while leaving at least 6p − 9 ≥ 2p − 2 variables of type E 0 0 unused. Lemma 32 can be used to contract 2p − 2 of the remaining E 0 0 variables together with p 2 − p of the C 0 variables to p − 1 variables of type C 1 . One can contract the p − 1 variables of type E 1 0 , the E 1 0 variable and the p − 1 variables of type C 1 to a C 2 variable, due to Lemma 22.
Case m 1 − I 1 0 ≥ 1. Use Lemma 25 to contract the E 0 0 variable to p − 2 variables of type E 1 0 while leaving at least 6p − 9 ≥ 2p − 2 unused. Then one can take Lemma 32 to contract p 2 − p of the C 0 variables together with 2p − 2 of the remaining E 0 0 variables to p − 1 variables of type C 1 . If I 1 0 ≥ 1, then one can use Lemma 22 to contract the p − 1 variables of type E 1 0 , the p − 1 variables of type C 1 and one of the E 1 0 variables to a C 2 variable. Thus, one can assume, that I 1 0 = 0, m 1 − I 1 0 ≥ 2 and If none of the C 1 variable is already of type C 2 , they are all E 1 variables. Take a set K containing the C 1 variables, two of the E 1 0 variables which exist due to m 1 − I 1 0 ≥ 2 and the p − 2 variables of type E 1 0 . If there is a µ such that I µ (K ) ≥ p, then there is at least one C 1 µ variable in K . Due to Lemma 19 one can contract the variables in K of colour µ to an C 2 variable. Else, one has q (K ) ≥ p, because K = 2p − 1. It follows, that one has transformed the pair f, g into a pair with m 1 ≥ 2p − 1 and q 1 ≥ p. The new pair either has a non-trivial p-adic solution due to Lemma 36, from which it would follow that f, g has one as well, or it has q 2 ≤ p − 2 or I 2 max ≤ p − 2. As the new pair has the same parameter q 2 and I 2 max as the pair f, g, one can assume, that q 2 ≤ p − 2 or I 2 max ≤ p − 2 holds for f, g as well. This contradicts the p-normalisation, because then one of the inequalities holds, hence, it follows that this case cannot occur.
This concludes the case r = −1 and with that the claim follows.
This shows that for every proper p-normalised pair f, g the equations f = g = 0 have a non-trivial p-adic solution provided that τ = 1.

Pairs of Forms with τ ≥ 2
This section will prove the theorem for τ ≥ 2, which completes the proof. In general, the proof relies on the same techniques independent on the actual value of τ , but sometimes one has to separate the cases τ = 2 and τ = 3, because the proof is easier for bigger τ and hence, the cases τ ∈ {2, 3} require some extra effort.
In order to avoid a repetition of the same argument, the following lemma will point out a situation in which one can contract a C τ +1 or a P τ +1 variable, which will appear constantly in the proof for τ ≥ 2.
Proof. One can contract the variables of type S j and type E i νi for i ∈ {j, . . . , τ − 1} to p + m variables at level of type S τ due to Lemma 32. Those and the p − m − 1 variables of type E τ ν can be contracted to a variable of type S τ +1 using Lemma 21.
The following lemma focuses on cases, where the number of variables at level 0 is small.
Then the equations f = g = 0 have a non-trivial p-adic solution.
Proof. By the p-normalisation of f, g, one has q 0 ≥ p τ +1 − p τ + 1 and m 0 ≥ 2p τ +1 − 2p τ + 1, from which it follows that one can contract the variables at level 0 to p τ − p τ −1 variables of type P 1 due to Lemma 23. The upper bound of m 0 provides the bounds Therefore, there are at least ν for all 0 ≤ ν ≤ p. Those variables can be contracted together with the P 1 variables to 2p τ −1 + p τ −2 − 2 variables of type P 2 by using Lemma i=0 p i − 1 and z = p − 2. Then Lemma 31 can be used to contract the P 2 variables to 2p − 1 variables of type P τ , which contract to a P τ +1 variable due to Lemma 14. For bigger m 0 it will be helpful to divide the cases depending on the value of r (f, g). The following three lemmata will complete the proof that a for a proper p-normalised pair f, g with τ ≥ 2 and r = r (f, g) ≥ 0 the equations f = g = 0 have a non-trivial p-adic solution.
This will be done by using different strategies depending on the size of m 0 . The area of the value of m 0 in which one has to use a certain strategy differs between p ≥ 7 and p = 5. This is due to some inequalities, which do not hold if p is too small. To counter this, the lemmata that are stronger in the case p = 5 will be used, which results in the different areas.
variables of type E 1 0 with Lemma 26 for p = 5. This leaves 6p − 9 ≥ 2p − 2 variables of type E 0 0 unused in both cases. If on the other hand, one has ν = 0, it follows that and by m 1 − I 1 0 = q 1 and (5.1) that Thus one can contract p τ +1 + p 2 − 2p − q 1 p of the E 0 0 variables to p τ + p − 2 − q 1 variables of type E 1 0 due to Lemma 30, leaving at least p τ +1 − 3p τ − p 2 + 2p + 2 + (p − 1) q 1 ≥ 2p − 2 variables of type E 0 0 unused. In both cases, one has contracted enough E 1 ν variables to have at least p τ + p − 2 variables of type E 1 ν , while there are 2p−2 variables of type E 0 0 remaining. The E 0 0 variables can be contracted together with p τ +1 − p τ of the C 0 variables to p τ − p τ −1 variables of type C 1 , using Lemma 32. Then, one can contract 4p i=0 p i + 1 and z = p − 2, to 2p τ −1 − p τ −2 variables of type C 2 . With Lemma 39 those and the 2p − 2 variables in E j ν for j ∈ {2, . . . , τ − 1} can be contracted to a C τ +1 variable. Thus, from now on, one can assume, that By Lemma 32, the p τ +1 − p τ variables of type C 0 can be contracted together with the 2p − 2 variables of type E 0 0 to p τ − p τ −1 variables of type C 1 . Using Lemma 29 with x = p τ −1 − 2p τ −2 − ∑ τ −3 i=0 p i − 1, y = ∑ τ −1 i=0 p i + 1 and z = p − 2, one can contract the E 1 variables together with the C 1 variables to 2p τ −1 − p τ −2 variables of type C 2 , which contract together with the 2p − 2 variables of type E i 0 for i ∈ {2, . . . , τ − 1} to a C τ +1 variables due to Lemma 39. Therefore, one can assume that either m 1 ≤ 2p τ + p 2 − p − 4 or q 1 ≤ p τ + p − 3. The latter case leads to m 1 = q 1 + I 1 i=0 p i − 8 due to (8.5). Hence, from now on, one can assume that p i + p 2 − 8, (8.6) because this is an upper bound for the upper bound for m 1 in both cases.