Consequences of the Zero Statements

In this section we discuss resulting properties which can be deduced from the preceding section on real zeros of SONCs.

An important question for SONC polynomials (resp. forms) is whether the analog to Hilbert's 17th problem is true, see Section 2.2. That is, if every nonnegative polynomial is representable as a nite sum of nonnegative circuit polynomials of rational functions, i.e., a sum of quotients of nonnegative circuit polynomials. By means of a zero argument we are able to answer this analog question of Hilbert for SONC polynomials in the negative.

Corollary 3.2.12. Let f P P_n,2d. Then, in general, there are no nonnegative circuit polynomials g1, . . . , gr, h1, . . . , hr PCn,2d, and hj 0 for j 1, . . . , r, such that

f

¸r j1

g_j h_j

.

Proof. We prove the assertion by inspecting the zero set of the left hand side and the right hand side of the equation. By Corollary 3.2.4 it holds |Vp°

jgjq| ¤ 2ⁿ, thus

|Vp°

jp_h^gj_jqq| ¤2ⁿ.

For all n and d there exist polynomials f P P_n,2d such that |Vpfq| ¥ dⁿ, see [CLR80, Proposition 4.1]. Hence, we have |Vpfq| ¥ dⁿ¡2ⁿ, for d¥3.

Note that by Corollary 3.2.9 and the subsequent Theorem 3.2.14, the analog of Hilbert's 17th problem cannot be true in general in the homogeneous case as well.

As we will see in Lemma 3.4.1 the set of SONC polynomials is not closed under multiplication. Therefore, the question, if there always exists a suitable multiplier for a nonnegative polynomial to be a sum of nonnegative circuit polynomials is not equivalent to Corollary 3.2.12. However, with similar arguments as above one can show, that the multiplier question can also be answered in the negative.

In [CLR80] Choi, Lam, and Reznick considered the numbers B_{n 1,2d} and B_n¹ _1,2d, where B_{n 1,2d} (resp. B_n¹ _1,2d) is dened as sup|Vppq|, wherep ranges over all forms in P_{n 1,2d} (resp. in Σ_{n 1,2d}) with sup|Vppq|   8, see also Section 2.2.3. They noticed that the determination of these numbers is quite challenging and presented some partial results. Moreover, they observed that for generaln and dit is unclear if B_{n 1,2d} always needs to be nite. See also Theorem 2.2.26 for results in special cases.

Inspired by this, we dene an analog number for SONC forms.

Denition 3.2.13.

B_n² _1,2d: sup

pPCn 1,2d

|Vppq| 8

|Vppq|.

7 A crucial dierence to the numbers B_{n 1,2d} and B_n¹ _1,2d is that in our case such a number B_n² _1,2d is always nite and actually can be given explicitly.

Theorem 3.2.14. Let B²_{n 1,2d} be dened as above, then:

(1) Special case d1 : B_2,2² 1.

(2) B_2,4² 2, B_2,6² 3, and if 2d¥8 we have B_2,2d² 4. (3) B_3,4² 3 and B_3,2d² 7 for 2d¥6.

(4) For all n 1 ¥ 4 : B_n² _1,2d 2^2d2 3 for 2d   n 1, B_n² _1,2d 2ⁿ¹ 3 if n 1¤2d 2pn 1q, and B_n² _1,2d2ⁿ 3 for 2pn 1q ¤ 2d.

Proof.

(1) For d 1 we have a special case, since the only possibility for a proper SONC form of degree 2 is the circuit form f f_αp0qx²₀ f_αp1qx²₁Θ_fx₀x₁, which has only one zero r1 : 1s. Even if we consider a sum of monomial squares, the only zero in the case of degree2 would be r0 : 0s R P².

(2) First, note that the maximum number of zeros of a sum of monomial squares in the case n 1 2 is 2, namely the single monomial square x²₀x²₁ has the zeros r0 : 1s and r1 : 0s. If we consider proper SONC forms, then the number of zeros depends on the degree, because certain vertex constellations are only possible from a certain degree on. For 2d 4 we have, up to renumbering of the variables, only two possible circuit forms f₁ f_αp0qx⁴₀ f_αp1qx⁴₁ Θ_f1x²₀x²₁ with zeros r1 : 1s,r1 : 1s and f₂ f_αp0qx²₀x²₁ f_αp1qx⁴₀ Θ_f2x³₀x₁ with zeros r1 : 1s,r0 : 1s. Therefore, the rst assertion in (2) holds. The second follows by the observation that for 2d 6 there exists a circuit form the rst time, for which one outer term consists of both variables and the inner term has an even exponent: f f_αp0qx²₀x⁴₁ f_αp1qx⁶₀Θ_fx⁴₀x²₁. This gives the zeros r1 : 1s,r1 :1s, and r0 : 1s. Lastly, if the degree is greater or equal than 8, a circuit form with even inner exponent exists, for which both outer terms consist of both variables:

f f_αp0qx²₀x⁶₁ f_αp1qx⁶₀x²₁ Θ_fx⁴₀x⁴₁. It has the four zeros r1 : 1s,r1 : 1s,r0 : 1s, and r1 : 0s. Obviously, a bivariate SONC form cannot have more than 4zeros.

(3) Observe that in the case of n 1 3 the maximum number of zeros by a sum of monomial squares is3. More precisely consider the following sum of monomial squares without loss of generality in degree4: m x²₀x²₁ x²₁x²₂ x²₀x²₂. Obviously, r1 : 0 : 0s,r0 : 1 : 0s, and r0 : 0 : 1s are the zeros ofm.

For reasons of realizability, see Lemma 3.1.4, a proper circuit form of degree4has an odd inner exponent, and for2d ¥6, also a proper circuit form with even inner exponent is possible. Thus, the statements follow immediately by Corollary 3.2.9 (1) and the preliminary consideration.

(4) The last two assertions are a direct result of Lemma 3.1.4 and Corollary 3.2.9 (2).

Note that as forn 13, the maximum number of zeros by a sum of monomial squares is 3. In the case 2d n 1 there exists no proper circuit form. Though there are SONC forms p, which consist of a sum of a proper circuit form f and a sum of monomial squares. We know that a proper circuit form with odd inner exponent exists if the number of variables is equal to2d. Hence, if2d n 1we

have the following SONC form:

pf x^2d_{2d 1} x^2d_n ,

wheref is a2d-variate proper circuit form. Thus, |Vppq| |Vpfq|, which leads to the equality B_n² _1,2d B_2d,2d² . By Corollary 3.2.9 (2), with n 1 2d, it follows B_2d,2d² 2^p2d1q1 3.

The following example serves to illustrate the considerations of the case (4) for 2d n 1 in the proof above.

Example 3.2.15. We want to verify the calculationB_7,4² 2⁴² 37. LetpP BP_7,4 be a SONC polynomial. Obviously4 7, therefore we search a proper4-variate circuit form. Consider for instance

f 1

4x²₀x²₁ 1

4x²₁x²₂ 1

4x²₂x²₀ 1

4x⁴₃x₀x₁x₂x₃. The zero set of this form is

Vpfq tr1 : 1 : 1 : 1s,r1 : 1 :1 :1s,r1 :1 : 1 : 1s,r1 : 1 : 1 :1s, r1 : 0 : 0 : 0s,r0 : 1 : 0 : 0s,r0 : 0 : 1 : 0su.

Hence, |Vpfq| 7. Thus, the SONC formp, p 1

4x²₀x²₁ 1

4x²₁x²₂ 1

4x²₂x²₀ 1

4x⁴₃x₀x₁x₂x₃ x⁴₄ x⁴₅ x⁴₆,

has the same number of zeros as f, namely 7. 7

We conclude this section with a short comparison of the“B-numbers of the dierent cones Pn 1,2d,Σn 1,2d, and Cn 1,2d.

Remark 3.2.16.

(i) First note that B_2,2 B_2,2¹ B_2,2² 1, which is in line with the fact, that for pn 1,2dq p2,2q the three cones coincide, see Theorem 3.1.2.

(ii) In the bivariate case one has B_2,2d B_2,2d¹ d, which equals B_2,2d² for 2d ¤ 8. Therefore, we have a rst dierence in the number of real zeros for degree 10.

(iii) In the case n 13 we have the following observations, B_3,4 B_3,4¹ B_3,4² 4. But from degree 6 on, there are dierences in the numbers of zeros: B3,6 10, B_3,6¹ 9, and B_3,6² 7.

(iv) Finally, we take a look atn 14. Here, we already have dierences for quartics:

B_4,4 10, B_4,4¹ 8, andB_4,4² 7.

3.3 Exposed Faces of the SONC Cone in Small