In order to provide a second-order representation for the \({\mathcal {S}}\)-cone and its dual, the main task is to capture the cone of non-negative AG functions and its dual. For a comprehensive collection of techniques for handling second-order cones, we refer to Ben-Tal and Nemirovski (2001).
Throughout the section, let \((A,\beta )\) be a fixed circuit and rational barycentric coordinates \(\lambda \in {\mathbb {R}}_+^A\), which represent \(\beta \) as a convex combination of A. That is, \(\beta = \sum _{\alpha \in {\mathcal {A}}} \lambda _{\alpha } \alpha \) and \(\sum _{\alpha \in {\mathcal {A}}} \lambda _{\alpha } =1\). Let \(p\in {\mathbb {N}}\) denote the smallest common denominator of the fractions \(\lambda _\alpha \) for \(\alpha \in A\), i.e., \(\lambda _\alpha =\frac{p_\alpha }{p}\) with \(p_\alpha \in {\mathbb {N}}\) for all \(\alpha \in A\) and p is minimal.
With the given circuit \((A,\beta )\in I({\mathcal {A}})\), we associate a set of dual circuit variables
$$\begin{aligned} (y_{k,i})_{k,i}, \end{aligned}$$
(3.1)
where \(k\in [\lceil \log _2(p)\rceil -1]\) and \(i\in [2^{\lceil \log _2(p)\rceil -k}]\). The collection of these \(\sum _{k=1}^{\lceil \log _2(p)\rceil -1} 2^{\lceil \log _2(p)\rceil -k} \) \(= 2^{\lceil \log _2(p)\rceil }-2\) variables is denoted as \({\mathbf {y}}^{A,\beta }\) or shortly as \({\mathbf {y}}\). Further, denote the restriction of a vector \({\mathbf {v}}\in {\mathbb {R}}^{{\mathcal {A}}}\) to the components of \(A \subseteq {\mathcal {A}}\) by \({\mathbf {v}}_{|A}\).
Definition 3.1
A dual circuit matrix \(C_{A,\beta }^*({\mathbf {v}}_{| A},v_\beta ,{\mathbf {y}})\) is a block diagonal matrix consisting of the blocks
$$\begin{aligned}&\left( \begin{array}{cc} y_{k-1,2i-1} &{} y_{k,i} \\ y_{k,i} &{} y_{k-1,2i} \end{array}\right) \quad \text {for }k\in \{2,\ldots ,\lceil \log _2(p)\rceil -1 \}\text { and } i\in \left[ 2^{\lceil \log _2(p)\rceil -k}\right] , \end{aligned}$$
(3.2)
$$\begin{aligned}&\left( \begin{array}{cc} y_{\lceil \log _2(p)\rceil -1,1} &{} v_\beta \\ v_\beta &{} y_{\lceil \log _2(p)\rceil -1,2} \end{array}\right) , \end{aligned}$$
(3.3)
the singleton block \( ( v_\beta ), \) as well as \(2^{\lceil \log _2(p)\rceil -1}\) blocks of the form
$$\begin{aligned} \left( \begin{array}{cc} u &{} y_{1,l} \\ y_{1,l} &{} w \end{array}\right) \quad \text { for } l\in [2^{\lceil \log _2(p)\rceil -1} ], \end{aligned}$$
(3.4)
where in each of these blocks u and w represent a variable of the set \(\{v_\alpha \, : \, \alpha \in A\}\cup \{v_\beta \}\) such that altogether each \(v_{\alpha }\) appears \(p_{\alpha }\) times and \(v_\beta \) appears \(2^{\lceil \log _2(p)\rceil }-p\) times.
In this definition, the exact order of appearances of the variables in \(\{v_\alpha \, : \, \alpha \in A\}\cup \{v_\beta \}\) is not uniquely determined. However, since this order of appearances will not matter, we will speak of the dual circuit matrix.
Remark 3.2
Each block of the type (3.4) contains two (not necessarily identical) variables from the set \(\{v_\alpha \, : \, \alpha \in A\}\cup \{v_\beta \}\). Since \(\sum _{\alpha \in A} \lambda _{\alpha } = 1\), we have \(\sum _{\alpha \in A} p_{\alpha } = p\) and hence the total number of occurrences of variables from the set \(\{v_\alpha \, : \, \alpha \in A\}\cup \{v_\beta \}\) in the blocks of type (3.4) is
$$\begin{aligned} \sum _{\alpha \in A} p_{\alpha } + (2^{\lceil \log _2(p)\rceil }-p) = 2^{\lceil \log _2(p)\rceil }, \end{aligned}$$
which is twice the number of blocks of type (3.4).
Note that every \(y_{k,i}\) only serves as an auxiliary variable to make the non-linear constraints \( \ln (v_\beta ) \le \sum \nolimits _{\alpha \in A} \lambda _\alpha \ln (v_\alpha ) \) of the dual \({\mathcal {S}}\)-cone description from Proposition 2.4 linear. In the end, we will only multiply those constraints to obtain the original ones. In particular, factors \(v_\beta \) serve to cover cases where p is not a power of 2. For the purpose of the second-order descriptions, it does not matter in which order the variables appear in the blocks (3.4), because only the product of these blocks will be considered.
The goal of this subsection is to show the following characterization of the cone of non-negative even AG functions \(P^{\mathrm {even}}_{A,\beta }\) supported on the circuit \((A,\beta )\). Here, positive semidefiniteness of a symmetric matrix is denoted by \(\succeq 0\).
Theorem 3.3
The dual cone \((P^{\mathrm {even}}_{A,\beta })^*\) of the cone of non-negative even AG functions \(P^{\mathrm {even}}_{A,\beta }\) supported on the circuit \((A,\beta )\in I({\mathcal {A}})\) is the projection of the spectrahedron
$$\begin{aligned}&\left\{ ({\mathbf {v}}, {\mathbf {y}}) \in {\mathbb {R}}^{{\mathcal {A}}}\times {\mathbb {R}}^{2^{\lceil \log _2(p)\rceil }-2} \ : \ C_{A,\beta }^*({\mathbf {v}}_{| A},v_\beta , {\mathbf {y}}) \succcurlyeq 0 \right\} \end{aligned}$$
(3.5)
on \(({\mathbf {v}}_{|A}, v_\beta )\). \((P^{\mathrm {even}}_{A,\beta })^*\) is second-order representable.
Here, the second-order representability follows immediately from the representation (3.5) in connection with Lemma 2.5. Let us consider an example for the theorem.
Example 3.4
Let \({\mathcal {A}}=\{0,6\}, {\mathcal {B}}=\{2\}\) and consider the circuit \((A,\beta )\) with \(A={\mathcal {A}}\) and \(\beta =2\) (compare Fig. 1). We have \(p=3, p_0=2, p_6=1\) and \({\mathbf {y}}\) consists of the components
$$\begin{aligned} \ y_{1,1}, \ y_{1,2}. \end{aligned}$$
A vector \((v_0,v_2,v_6)\) is contained in \((P^{\mathrm {even}}_{A,\beta })^*\) if and only if \(v_2\ge 0\) and the three \(2 \times 2\)-matrices
$$\begin{aligned} \left( \begin{array}{cc} y_{1,1} &{} v_2 \\ v_2 &{} y_{1,2} \end{array} \right) , \; \left( \begin{array}{cc} v_0 &{} y_{1,1} \\ y_{1,1} &{} v_0 \end{array} \right) , \; \left( \begin{array}{cc} v_6 &{} y_{1,2} \\ y_{1,2} &{} v_2 \end{array}\right) \end{aligned}$$
are positive semidefinite.
In Averkov (2019), Averkov considered the size of the blocks in the SDP-representation of SONC-polynomials but does not give a number or bound on the number of blocks. Here, for the \({\mathcal {S}}\)-cone, we provide a bound on the number of inequalities of a second-order representation, which also gives a bound on the number of \(2 \times 2\)-blocks in a semidefinite representation. The bound depends on the smallest common denominator of the barycentric coordinates representing the inner exponent of a circuit as a convex combination of the outer ones.
Corollary 3.5
The matrix \(C_{A,\beta }^*({\mathbf {v}}_{|A},v_\beta ,{\mathbf {y}})\) consists of \({2^{\lceil \log _2(p)\rceil }}-1\) blocks of size \(2 \times 2\) and one block of size \(1 \times 1\).
Proof
Counting the number of \(2 \times 2\)-blocks, there are \(\sum _{k=2}^{\lceil \log _2(p)\rceil -1}\left( 2^{\lceil \log _2(p)\rceil -k} \right) = 2^{\lceil \log _2(p)\rceil -1}\) \(-2\) blocks of type (3.2), a single block (3.3) and \(2^{\lceil \log _2(p)\rceil -1}\) blocks of type (3.4). \(\square \)
Remark 3.6
It is useful to record the set inequalities characterizing the positive semidefiniteness of the matrix \(C_{A,\beta }^*({\mathbf {v}}_{|A},v_\beta ,{\mathbf {y}})\). Besides the non-negativity conditions for the variables,
$$\begin{aligned}&{\mathbf {v}}_{|A} \ge 0, \quad v_\beta \ge 0, \end{aligned}$$
(3.6)
$$\begin{aligned}&\quad \text { and } x_{k,i} \ge 0 \text { for all } k\in \left\{ 2,\ldots ,\lceil \log _2(p)\rceil -1\right\} , i\in \left[ 2^{\lceil \log _2(p)\rceil }-k\right] , \end{aligned}$$
(3.7)
these are the determinantal conditions arising from the positive semidefiniteness of the matrices in (3.2), (3.3) and (3.4):
$$\begin{aligned} v_\beta ^2&\le {y_{\lceil \log _2(p)\rceil -1,1}y_{\lceil \log _2(p)\rceil -1,2}}, \end{aligned}$$
(3.8)
$$\begin{aligned} y_{k,i}^2&\le y_{k-1,2i-1} y_{k-1,2i} \text { for all }k\in \left\{ 2,\ldots ,\lceil \log _2(p)\rceil -1\right\} , i\in \left[ 2^{\lceil \log _2(p)\rceil -k}\right] \end{aligned}$$
(3.9)
$$\begin{aligned}&\text { and } uw \ge \left( y_{1,l}\right) ^2 \text { for } l\in \left[ 2^{\lceil \log _2(p)\rceil -1}\right] \end{aligned}$$
(3.10)
for \(u,w\in \{v_\alpha \, : \, \alpha \in A\}\cup \{v_\beta \}\), such that \(v_{\alpha }\) appears \(p_{\alpha }\) times for every \(\alpha \in A\) and \(v_\beta \) appears \(2^{\lceil \log _2(p)\rceil }-p\) times.
The next lemma prepares one inclusion of Theorem 3.3.
Lemma 3.7
Let \({\mathbf {v}}\in {\mathbb {R}}^{A,\beta }\) such that there exists \({\mathbf {y}}\in {\mathbb {R}}^{2^{\lceil \log _2(p)\rceil }-2}\) with \(C_{A,\beta }^*({\mathbf {v}}_{|A},v_\beta , {\mathbf {y}}) \succcurlyeq 0\). Then \({\mathbf {v}}_{|A}\) is non-negative and satisfies
$$\begin{aligned} v_\beta ^p \le \prod \limits _{\alpha \in A} v_\alpha ^{p_\alpha }. \end{aligned}$$
Proof
By (3.6), we have \({\mathbf {v}}_{|A} \ge 0\) and \(v_{\beta } \ge 0\). Moreover, (3.8) and successively applying (3.9) gives
$$\begin{aligned} v_\beta\le & {} \left( y_{\lceil \log _2(p)\rceil -1,1} \, y_{\lceil \log _2(p)\rceil -1,2}\right) ^{1/2} \\\le & {} \left( y_{\lceil \log _2(p)\rceil -2,1} \, y_{\lceil \log _2(p)\rceil -2,2} \right) ^{1/4} \left( y_{\lceil \log _2(p)\rceil -2,3} \, y_{\lceil \log _2(p)\rceil -2,4}\right) ^{1/4} \\= & {} \left( y_{\lceil \log _2(p)\rceil -2,1} \, y_{\lceil \log _2(p)\rceil -2,2} \, y_{\lceil \log _2(p)\rceil -2,3} \, y_{\lceil \log _2(p)\rceil -2,4}\right) ^{\frac{1}{2^{\lceil \log _2(p)\rceil -(\lceil \log _2(p)\rceil -2)}}}\\\le & {} \cdots \le \left( \left( \prod \nolimits _{\alpha \in A} v_\alpha ^{p_\alpha }\right) \cdot \left( v_\beta \right) ^{2^{\lceil \log _2(p)\rceil }-p} \right) ^{\frac{1}{2^{\lceil \log _2(p)\rceil }}}. \end{aligned}$$
This is equivalent to
$$\begin{aligned} \left( v_\beta \right) ^{2^{\lceil \log _2(p)\rceil }} \cdot \left( v_\beta \right) ^{p-2^{\lceil \log _2(p)\rceil }} \le \prod \nolimits _{\alpha \in A} v_\alpha ^{p_\alpha }, \end{aligned}$$
which implies \( v_\beta ^p \le \prod _{\alpha \in A} v_\alpha ^{p_\alpha }. \) \(\square \)
Now we prepare the converse inclusion of Theorem 3.3.
Lemma 3.8
For every \({\mathbf {v}}\in {\mathbb {R}}^{A,\beta }\) with \({\mathbf {v}}_{|A \cup \{\beta \}} \ge 0\) and \(v_\beta ^p \le \prod _{\alpha \in A} v_\alpha ^{p_\alpha }\), there exists \({\mathbf {y}}\in {\mathbb {R}}^{2^{\lceil \log _2(p)\rceil }-2}\) such that \(C_{A,\beta }^*({\mathbf {v}}_{|A},v_\beta , {\mathbf {y}}) \succcurlyeq 0\).
Proof
Define \({\mathbf {y}}\) inductively by
$$\begin{aligned}&y_{1,l}=\sqrt{uw} \text { for those } u,w \text { which occur in the block with }y_{1,l}, \\&y_{k,i}=\sqrt{y_{k-1,2i-1}y_{k-1,2i}} \text { for all }k\in \left\{ 2,\ldots ,\lceil \log _2(p)\rceil -1\right\} , i\in \left[ 2^{\lceil \log _2(p)\rceil -k}\right] . \end{aligned}$$
It suffices to show that the inequalities (3.6)–(3.10) in Remark 3.6 are satisfied. The non-negativity conditions (3.6) and (3.7) hold by assumption and by definition of \({\mathbf {y}}\). The construction of \({\mathbf {y}}\) also implies that a subchain of the chain of inequalities considered in the previous proof even holds with equality,
$$\begin{aligned}&\left( y_{\lceil \log _2(p)\rceil -1,1} \, y_{\lceil \log _2(p)\rceil -1,2}\right) ^{1/2} \\&\quad = \left( y_{\lceil \log _2(p)\rceil -2,1} \, y_{\lceil \log _2(p)\rceil -2,2} \right) ^{1/4} \left( y_{\lceil \log _2(p)\rceil -2,3} \, y_{\lceil \log _2(p)\rceil -2,4}\right) ^{1/4} \\&\quad = \left( y_{\lceil \log _2(p)\rceil -2,1} \, y_{\lceil \log _2(p)\rceil -2,2} \, y_{\lceil \log _2(p)\rceil -2,3} \, y_{\lceil \log _2(p)\rceil -2,4}\right) ^{\frac{1}{2^{\lceil \log _2(p)\rceil -(\lceil \log _2(p)\rceil -2)}}}\\&\quad = \cdots = \left( \left( \prod \nolimits _{\alpha \in A} v_\alpha ^{p_\alpha }\right) \cdot \left( v_\beta \right) ^{2^{\lceil \log _2(p)\rceil }-p} \right) ^{\frac{1}{2^{\lceil \log _2(p)\rceil }}}. \end{aligned}$$
By the assumption \(v_\beta ^p \le \prod _{\alpha \in A} v_\alpha ^{p_\alpha }\), we obtain \(v_\beta ^2 \le {y_{\lceil \log _2(p)\rceil -1,1}y_{\lceil \log _2(p)\rceil -1,2}}\), which shows inequality (3.8). The remaining inequalities (3.9), (3.10) are satisfied with equality by construction. \(\square \)
Finally, we can conclude the proof of Theorem 3.3.
Proof of Theorem 3.3
Let p be defined as in Definition 3.1 and \(\lambda \in {\mathbb {R}}^A\) denote the barycentric coordinates representing \(\beta \) as a convex combination of A, i.e., \(\lambda _\alpha =\frac{p_\alpha }{p}\) with \(p_\alpha \in {\mathbb {N}}\) for all \(\alpha \in A\). By (2.3) and Proposition 2.4, we have
$$\begin{aligned} (P^{\mathrm {even}}_{A,\beta })^*&=\left\{ {\mathbf {v}}\in {\mathbb {R}}^{A,\beta } \, : \, {\mathbf {v}}_{|A\cup \{\beta \}} \ge 0, \;\ln (v_\beta )\le \sum \nolimits _{\alpha \in A} \lambda _\alpha \ln (v_\alpha )\right\} \\&= \left\{ {\mathbf {v}}\in {\mathbb {R}}^{A,\beta } \, : \, {\mathbf {v}}_{|A\cup \{\beta \}}\ge 0, \; v_\beta ^p \le \prod \nolimits _{\alpha \in A} v_\alpha ^{p_\alpha }\right\} . \end{aligned}$$
Applying Lemmas 3.7 and 3.8, we obtain that \(C^*_{A,\beta }(x,v_\beta )\succcurlyeq 0\) if and only if \({\mathbf {v}}\in P_{A,\beta }^*\). \(\square \)
Our derivation of the second-order representation of the dual cone \((P^{\mathrm {even}}_{A,\beta })^*\) also suggests a simple way to derive a second-order cone representation of the primal cone \(P^{\mathrm {even}}_{A,\beta }\). For the dual cone, Proposition 2.4 gives—besides non-negativity-constraints on \(v_\alpha \) for \(\alpha \in {\mathcal {A}}\) and on \(v_{\beta }\)—the condition \( \ln (v_\beta ) \le \sum \nolimits _{\alpha \in A} \lambda _\alpha \ln (v_\alpha ) \) for every circuit \((A,\beta )\in I({\mathcal {A}})\). Those conditions can—as done in the previous proof – be stated as
$$\begin{aligned} v_\beta ^p \le \prod _{\alpha \in A} v_\alpha ^{p_\alpha }, \text { where } \lambda _\alpha =\frac{p_\alpha }{p}. \end{aligned}$$
The conditions for the primal cone can be reformulated similarly. Namely, by (2.6), an even circuit function f with coefficient vector \({\mathbf {c}}\) is non-negative if and only if \( -c_{\beta } \le \prod _{\alpha \in A} \left( c_{\alpha } / \lambda _{\alpha } \right) ^{\lambda _{\alpha }}, \) which we write as
$$\begin{aligned} (-c_\beta )^p \le \prod _{\alpha \in A} \left( \frac{c_\alpha }{\lambda _\alpha }\right) ^{p_\alpha }. \end{aligned}$$
This motivates to carry over the definition of the dual circuit matrix to the primal case as follows. Since \(c_\beta \) may be negative (in contrast to the dual case), we introduce the primal circuit variables, or simply circuit variables,
$$\begin{aligned} (x_\beta , (x_{k,i})_{k,i}), \end{aligned}$$
where \(k\in [\lceil \log _2(p)\rceil ]\) and \(i\in [2^{\lceil \log _2(p)\rceil -k}]\). As in the dual case, we refer to these \(1+\sum _{k=1}^{\lceil \log _2(p)\rceil }2^{\lceil \log _2(p)\rceil -k}=2^{\lceil \log _2(p)\rceil }\) variables as \({\mathbf {x}}^{A,\beta }\) or shortly as \({\mathbf {x}}\).
Definition 3.9
(Circuit matrix) The circuit matrix \(C_{A,\beta }({\mathbf {c}}_{|A\cup \{\beta \}},x_\beta ,{\mathbf {x}})\) is the block diagonal matrix consisting of the blocks
$$\begin{aligned}&\left( \begin{array}{cc} x_{k-1,2i-1} &{} x_{k,i} \\ x_{k,i} &{} x_{k-1,2i} \end{array}\right) \quad \text { for } k\in \left\{ 2,\ldots , \lceil \log _2(p)\rceil \right\} , \ i\in \left[ 2^{\lceil \log _2(p)\rceil -k}\right] , \end{aligned}$$
the two singleton blocks
$$\begin{aligned} \left( \begin{array}{c} x_{\lceil \log _2(p)\rceil ,1} - \left( \prod \nolimits _{\alpha \in A} (\lambda _\alpha )^{\lambda _\alpha }\right) x_\beta \end{array}\right) , \quad \left( \begin{array}{c} x_\beta +c_\beta \end{array}\right) , \end{aligned}$$
(3.11)
as well as \(2^{\lceil \log _2(p)\rceil -1}\) blocks of the form
$$\begin{aligned} \left( \begin{array}{cc} u &{} x_{1,l} \\ x_{1,l} &{} w \end{array}\right) \quad \text { for } l\in [2^{\lceil \log _2(p)\rceil -1} ], \end{aligned}$$
(3.12)
where \(u,w \in \{c_\alpha \, : \, \alpha \in A\}\cup \{\left( \prod \nolimits _{\alpha \in A} (\lambda _\alpha )^{\lambda _\alpha }\right) x_\beta \}\), such that \(c_{\alpha }\) appears \(p_{\alpha }\) times for every \(\alpha \in A\) and \(\left( \prod \nolimits _{\alpha \in A} (\lambda _\alpha )^{\lambda _\alpha }\right) x_\beta \) appears \(2^{\lceil \log _2(p)\rceil }-p\) times.
Note that for a circuit \((A,\beta )\), the product \(\left( \prod \nolimits _{\alpha \in A} (\lambda _\alpha )^{\lambda _\alpha }\right) \) is always non-zero, because \(\beta \in {{\,\mathrm{relint}\,}}{{\,\mathrm{conv}\,}}A\) and A consists of affinely independent vectors.
In contrast to the dual cone, there is no sign constraint on \(c_\beta \) in the primal cone. If p is not a power of 2, then \(x_\beta \) appears on the main diagonal of (3.12). In our coupling of \(x_\beta \) with \(c_{\beta }\), the constraint \(x_\beta +c_\beta \ge 0\) results in \(-c_\beta \le x_\beta \) and thus reflects these sign considerations.
Note that the primal cone consists of circuit functions, whereas in our definition of the dual cone, the elements are coefficient vectors. Therefore, the projection regarded in Theorem 3.3 only delivers the coefficients of the circuit functions rather than the cone itself.
Theorem 3.10
The set of coefficients of the cone \(P^{\mathrm {even}}_{A,\beta }\) of non-negative even circuit polynomials supported on the circuit \((A,\beta )\) coincides with the projection of the spectrahedron
$$\begin{aligned}&\widehat{P^{\mathrm {even}}_{A,\beta }} := \left\{ ({\mathbf {c}}, {\mathbf {x}}) \in {\mathbb {R}}^{{\mathcal {A}}}\times {\mathbb {R}}^{2^{\lceil \log _2(p)\rceil }} \, : \, C_{A,\beta }({\mathbf {c}}_{|A\cup \{\beta \}},x_\beta , {\mathbf {x}}) \succcurlyeq 0 , \ c_{|{\mathcal {A}}{\setminus } \left( A\cup \{\beta \}\right) }=0\right\} \end{aligned}$$
(3.13)
on \(({\mathbf {c}}_{|A},c_\beta )\). The cone \(P^{\mathrm {even}}_{A,\beta }\) is second-order representable.
The last equality constraint in (3.13) is redundant and can be omitted. We include it here, because this formulation is needed in Sect. 4 for the description of the \({\mathcal {S}}\)-cone supported on the full set \({\mathcal {A}}\).
Proof
First, let \(({\mathbf {c}},{\mathbf {x}})\in \widehat{P^{\mathrm {even}}_{A,\beta }}\). The positive semidefiniteness of the \(2\times 2\)-blocks in \(C_{A,\beta }({\mathbf {c}}_{|A\cup \{\beta \}}, \) \(x_\beta , {\mathbf {x}})\) imply the inequalities
$$\begin{aligned} {\mathbf {c}}_{|A} \ge 0 \text { and } (-x_\beta )^p \cdot \left( \prod \nolimits _{\alpha \in A} {\lambda _\alpha }^{\lambda _\alpha }\right) \le \prod \nolimits _{\alpha \in A} c_\alpha ^{p_\alpha }. \end{aligned}$$
The two \(1\times 1\)-blocks from (3.11) give the inequalities \( x_{\lceil \log _2(p)\rceil ,1} \ge \left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) x_\beta \text { and } x_\beta \ge -c_\beta . \) They imply \( -c_\beta \left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) \le x_\beta \left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) \le x_{\lceil \log _2(p)\rceil ,1}. \) Hence, similar to Lemma 3.7,
$$\begin{aligned}&x_\beta \left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) \le x_{\lceil \log _2(p)\rceil ,1}\ \le \ \left( x_{\lceil \log _2(p)\rceil -1,1} \, x_{\lceil \log _2(p)\rceil -1,2}\right) ^{1/2} \\&\quad \le \left( x_{\lceil \log _2(p)\rceil -2,1} \, x_{\lceil \log _2(p)\rceil -2,2} \right) ^{1/4} \left( x_{\lceil \log _2(p)\rceil -2,3} \, x_{\lceil \log _2(p)\rceil -2,4}\right) ^{1/4} \\&\quad = \left( x_{\lceil \log _2(p)\rceil -2,1} \, x_{\lceil \log _2(p)\rceil -2,2} \, x_{\lceil \log _2(p)\rceil -2,3} \, x_{\lceil \log _2(p)\rceil -2,4}\right) ^{\frac{1}{2^{\lceil \log _2(p)\rceil -(\lceil \log _2(p)\rceil -2)}}}\\&\quad \le \cdots \le \left( \left( \prod \nolimits _{\alpha \in A} c_\alpha ^{p_\alpha }\right) \cdot \left( x_\beta \right) ^{2^{\lceil \log _2(p)\rceil }-p}\left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) ^{2^{\lceil \log _2(p)\rceil }-p} \right) ^{\frac{1}{2^{\lceil \log _2(p)\rceil }}}. \end{aligned}$$
This is equivalent to
$$\begin{aligned} \left( x_\beta \right) ^{2^{\lceil \log _2(p)\rceil }}&\cdot \left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) ^{2^{\lceil \log _2(p)\rceil }} \cdot \left( x_\beta \right) ^{p-2^{\lceil \log _2(p)\rceil }}\\&\cdot \left( \prod \nolimits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha }\right) ^{p-2^{\lceil \log _2(p)\rceil }} \le \prod \nolimits _{\alpha \in A} c_\alpha ^{p_\alpha }, \end{aligned}$$
which, together with the considerations before the chain of inequalities, yields \( (-c_\beta )^p \le \prod _{\alpha \in A} (c_\alpha /\lambda _{\alpha })^{p_\alpha } \) and further \({\mathbf {c}}_{|A \cup \{\beta \}} \in P^{\mathrm {even}}_{A,\beta }\).
For the converse inclusion, we remind the reader that \(\lambda _\alpha >0\) for all \(\alpha \in A\). We set \(x_\beta := x_{\lceil \log _2(p)\rceil ,1}\left( \prod \nolimits _{\alpha \in A} \left( \frac{1}{\lambda _\alpha }\right) ^{\lambda _\alpha }\right) \) and, similar to the proof of Lemma 3.8, define \({\mathbf {x}}\) inductively by
$$\begin{aligned}&x_{1,l}=\sqrt{uw} \text { for those } u,w \text { which occur in the block with }x_{1,l}, \\&x_{k,i}=\sqrt{x_{k-1,2i-1}x_{k-1,2i}} \text { for all }k\in \{2,\ldots ,\lceil \log _2(p)\rceil \}, i\in \left[ 2^{\lceil \log _2(p)\rceil -k}\right] . \end{aligned}$$
Analogous to that proof, the construction of \({\mathbf {x}}\) gives \(C_{A,\beta }({\mathbf {c}}_{A \cup \{\beta \}},x_{\beta },{\mathbf {x}}) \succeq 0\).
Second-order representability is then an immediate consequence in view of Lemma 2.5. \(\square \)
Example 3.11
Let \({\mathcal {A}}=\{0,2\}\), \({\mathcal {B}}=\{1\}\) and consider the circuit \((A,\beta )\) with \(A={\mathcal {A}}\) and \(\beta =1\). Since
$$\begin{aligned} 1=\frac{1}{2}\cdot 0+ \frac{1}{2}\cdot 2, \end{aligned}$$
we have \(p_1=p_2=1\) and \(p=2\). Hence, \(\lceil \log _2(p) \rceil = \log _2(p)= 1\), \(2^{\lceil \log _2(p) \rceil }-p=2-p=0\) as well as
$$\begin{aligned} \prod \limits _{\alpha \in A} \lambda _\alpha ^{\lambda _\alpha } =\frac{1}{2} \; \text { and } \; {\mathbf {x}}=\left( \begin{array}{c} x_1 \\ x_{1,1} \end{array}\right) . \end{aligned}$$
A given vector \((c_0,c_1,c_2)\) is contained in \(P_{{\mathcal {A}},\beta }\) if and only if
$$\begin{aligned} x_{1,1} - \frac{1}{2}x_1 \ge 0, \; x_{1}+c_1 \ge 0 \; \text { and } \; \left( \begin{array}{cc} c_0 &{} x_{1,1} \\ x_{1,1} &{} c_2 \end{array}\right) \succeq 0. \end{aligned}$$
Similar to Lemma 3.5, we can determine the number of blocks.
Corollary 3.12
The matrix \(C_{A,\beta }({\mathbf {c}}_{|A\cup \{\beta \}},x_\beta ,{\mathbf {x}})\) consists of \({2^{\lceil \log _2(p)\rceil }}-1\) blocks of size \(2\times 2\) and two blocks of size \(1\times 1\).