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most recent 30 from 2019-02-21T17:01:12Z

Is this a possible strengthening of the Lehmer conjecture?

Tue, 11/06/2018 - 10:01

Here is another possible refinement of the Lehmer conjecture.

For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained in the field $\mathbb{Q}(\alpha)$. The discriminant of a minimal equation of $\alpha$ over a number field $F$ is an ideal of $O_F$ that we denote by $\Delta_F(\alpha)$. This is just the discriminant of the finite $O_F$-algebra $O_F[\alpha]$.

Let us write $d := [\mathbb{Q}(\alpha):\mathbb{Q}]$ and $d^{\mathrm{rel}} := [\mathbb{Q}(\alpha):C_{\alpha}] = [\mathbb{Q}^{\mathrm{ab}}(\alpha) : \mathbb{Q}^{\mathrm{ab}}]$, the absolute and the relative-over-cyclotomic degrees. The Amoroso-Dvornicich-Zannier relative Lehmer problem asks whether $h(\alpha) \gg 1 / d^{\mathrm{rel}}$ whenever $\alpha$ is not a root of unity. (This has been proved up to an $\epsilon$ in the exponent.) The Mahler discriminant inequality (an argument with the Vandermonde determinant and the Hadamard volume inequality) shows $$ |\Delta_{\mathbb{Q}}(\alpha)|^{1/d} \leq d M(\alpha)^2 $$ and $$ |N_{C_{\alpha}/\mathbb{Q}}\Delta_{C_{\alpha}}(\alpha)|^{1/d} \leq d^{\mathrm{rel}} M(\alpha)^2, $$ where $M(\alpha) := \exp(d h(\alpha))$ is the Mahler measure, and we define $$ \delta := \frac{d}{|\Delta_{\mathbb{Q}}(\alpha)|^{1/d} } \in [M(\alpha)^{-2},d], \quad \delta^{\mathrm{rel}} := \frac{d^{\mathrm{rel}}}{|N_{C_{\alpha}/\mathbb{Q}}\Delta_{C_{\alpha}}(\alpha)|^{1/d} } \in [M(\alpha)^{-2},d^{\mathrm{rel}}]. $$ Matveev's enhancement of Dobrowolski's theorem states $$ \log{M(\alpha)} \geq \Big( \frac{\log{\log{\max(\delta,3)}}}{\log{\max(\delta,3)}} \Big)^3 $$ (for all but finitely many algebraic numbers $\alpha$), just provided that the mild conditions $\mathbb{Q}(\alpha^p) = \mathbb{Q}(\alpha)$ hold for all primes $p < (\log{\delta})^2$. From this it is easy to see that the Lehmer conjecture holds true for all $\alpha$ having $\delta(\alpha)$ bounded from above: the case of large discriminants. Just note that, by a simple application of the Vahlen-Capelli theorem, $[\mathbb{Q}(\alpha) : \mathbb{Q}(\alpha^p)] > 1$ for a given prime $p$ entails the existence of an algebraic number $\eta \in \overline{\mathbb{Q}}^{\times}$ from the smaller field $\mathbb{Q}(\alpha^p)$, not a root of unity if $\alpha$ isn't, having a strictly lower degree $[\mathbb{Q}(\alpha^p):\mathbb{Q}]=d / [\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^p)]$ and with $M(\eta) = M(\alpha)$ and $\delta(\eta) \ll_p \delta(\alpha)$.

This last observation becomes a little more interesting once one realizes computationally that, among the algebraic integers of a Mahler measure $M(\alpha) < 2$ and a growing degree bound $D$, Matveev's defect $\delta(\alpha)$ appears to typically be bounded, say bounded with asymptotic probability $1$ as $D \to \infty$. But it may certainly go to infinity, as $\log\delta(\zeta_{N}) = \log{\log{d}} - \log{\log{\log{d}}} + O(1)$ when $N$ is a primorial level (the product of the first $k$ primes for some $k$), by an easy application of the prime number theorem.

But the argument in Matveev's paper yields a relative version of his result in which, crudely speaking, the left-hand side $\log{M(\alpha)}$ of the above lower bound is replaced by $$ \frac{\log{M(\alpha)}}{[C_{\alpha}:\mathbb{Q}]} = d^{\mathrm{rel}} h(\alpha), $$ the discriminant defect $\delta$ is replaced by its relative version $\delta^{\mathrm{rel}}$, and the exponent ``$3$'' on the right hand side is weakened to the slightly worse exponent $4$. By the same argument, the relative Lehmer conjecture (and a fortiori the original Lehmer conjecture) is true for all $\alpha$ having $\delta^{\mathrm{rel}}$ bounded above.

Thus I wanted to ask whether or not the latter could be anticipated as a possible refinement of the Lehmer conjecture:

Question. Is there any possibility to fancy an absolute constant upper bound on the relative discriminant defect $\delta^{\mathrm{rel}}$ of an algebraic number?

It is quite a strong statement, but I fail to see any way of refuting it, even heuristically. Of course, for the relative Lehmer problem, it is enough to restrict attention to the $\alpha$ with $h(\alpha) < 1 / d^{\mathrm{rel}}$. I am already interested in just a construction of any $\alpha$ having $\delta^{\mathrm{rel}}(\alpha) \to \infty$. The above cyclotomic examples are now suddenly rendered irrelevant in this relativized question. A special case is whether there is any hope to expect a $\log{|\Delta_{\mathbb{Q}}(\alpha)|} = d\log{d} - O(d)$ asymptotic on the logarithmic discriminant of a degree-$d$ algebraic number $\alpha$ with $M(\alpha) < 2$ for which the field $\mathbb{Q}(\alpha)$ does not include any roots of unity besides $\pm 1$.

English reference text on Snell envelopes of càdlàg processes w.o negative jumps

Tue, 11/06/2018 - 07:25

I am looking for an English reference on the theory of Snell envelopes of càdlàg processes with and without negative jumps. In particular which contain results on existence of Snell envelope and characterisation of optimal stopping times.

Karatzas book only treats nonnegative processes and the theorem on existence of an optimal stopping time assumes continuous processes.

Left lifting property and pushout

Tue, 11/06/2018 - 05:46

Let's say we are working in category $\mathcal{C}$, and that the three morphisms $ f: X \rightarrow X'$, $ g: Y \rightarrow Y'$ and $ h: Z \rightarrow Z'$ have the left lifting property with respect to a class of morphisms $S$.

Can we say that that the map $$ g \sqcup_{f} h : Y \sqcup_{X} Z \rightarrow Y' \sqcup_{X'} Z'$$

has also the left lifting lifting property with respect to $S$?

Given a lifting problem for the map $ g \sqcup_{f} h $ I can get lifting problems for the maps $f,g$ and $h$ but I cant "patch" them to get a map from the pushout solving my initial lifting problem...

Fundamental representations and weight space dimension

Mon, 11/05/2018 - 05:29

For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $1$-dimensional. If this is true, does it identify the fundamental representations, that is are the fundaental representation precisely those with $1$-dimensional weight spaces?

General SLLN-like asymptotic mean concentration

Mon, 11/05/2018 - 02:43

Disclaimer: As I am not very knowledgeable of the field to which this question pertains, I will introduce a temporary terminology to convey the idea of my question at the risk of conflicting with existing ones.

Let $(X_n)_{n\geq 0}$ be a sequence of random variables (not necessarily independent). Say such a sequence asymptotically concentrates (resp. $O$-concentrates) when

$$ X_n \stackrel{a.s.}{\sim} \mathbb{E}(X_n) \quad\text{(resp. } X_n \stackrel{a.s.}{\asymp} \mathbb{E}(X_n) \text{)}, $$ where the asymptotic is taken as $n\to +\infty$ and a.s. stands for almost surely (i.e. the probability of the associated limit is $1$)

An example of asymptotic concentration is $S_n := x_1 +\cdots + x_n$, where $x_n$ are independent boolean random variables, not necessarily identically distributed (Strong Law of Large Numbers: SLLN). Further instances of both the first and the second seem to be plenty in probabilistic combinatorics, though the one I am most familiar with is the Erdős-Tetali theorem (an example of $O$-concentration), where the $X_n$ are certain specific boolean polynomials (i.e. $f(x_1,\cdots,x_n)$ where $x_1,\cdots,x_n$ are independent boolean random variables).

As far as I gather, results on asymptotic mean concentration, as described, tend to be consequences of Hoeffding-type bounds and more concrete tail estimates and moment concentration inequalities (e.g. Janson's inequality, Kim-Vu inequality, and related estimates for boolean polynomials). In this sense, my question is:

Question: Is there a line of research directed at the "softer" problem of asymptotic concentration (and $O$-concentration) of families of certain types of r.v.s instead of the more concrete concentration inequalities? If so, where could I find more information about it?

Asymptotics of Littlewood polynomials

Sun, 11/04/2018 - 14:21

Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients. He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether there exists a sequence $f_n$ in $\mathscr F$ such that $$A\leq\frac{|f_n(\theta)|}{\sqrt{n+1}}\leq B$$ for large n's.

1. I'd like to know if this is still open and would be grateful if someone could provide references for recent survey papers.

2. If the conjecture is not settled yet, I was thinking if one can apply integer programming methods to tackle the problem?

Update 1 As pointed out by Fedor, J-P. Kahane's work gives an elegant proof for polynomials with unimodular coefficients. His result is as follows(I didn't translate it to English as seems rather easy to understand ):

Theoreme. Il existe une suite de polynomes $$P_n(z)=\sum_{m=1}^{n}a_{m,n}z^m,\,(|a_{m,n}| = 1; \, n=1,2,\ldots,m=1,\ldots,n)$$ et une suite $\epsilon_n$ positive tendant vers 0 telles que pour tout z de module 1 on ait$$(1-\epsilon_n)\sqrt{n}\leq|P_n(z)|\leq(1+\epsilon_n)\sqrt{n}.$$

Numerical experiments suggest there might be hope to construct a sequence of $P_n\in\mathbb {Z}_2[x]$) which satisfies Kahane's asymptotics.

[L]Littlewood, J. E., On polynomials $\sum^n\pm z^m, \sum^n e^{\alpha_mi}z^m, z = e^{\theta i}$, J. Lond. Math. Soc. 41, 367-376 (1966). ZBL0142.32603.

If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

Sun, 11/04/2018 - 10:52

I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+b=t$ is invertible then how can one verify the existence of certain $c$ and $c'$ in $R$ such that $ca+bc'=1$ (and so, $ca$ and $bc'$ are idempotents)?

It appears that this implication does not hold unconditionally; yet I would be deeply grateful for any hints that would allow to study it (note however that the case where $t-1$ is nilpotent is certainly simple). In this case I am interested in there is an extra conservativity assumption; in particular, in the quotient of $R$ by the two-sided ideal generated by $t-1$ non-invertible elements of $R$ do not become invertible. I have tried to relate my question with von Neumann regularity of elements of rings, but was not able to do this.

What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

Sun, 11/04/2018 - 02:54

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales:

...the single most important fact which distinguishes locales from spaces: the fact that every locale has a smallest dense sublocale. If you want to 'sell' locale theory to a classical topologist, it's a good idea to begin asking him to imagine a world in which any intersection of dense subspaces would always be dense. Once he has contemplated some of the wonderful consequences that would follow from this result you can tell him this world is exactly the category of locales.

Q. My classical topology in somewhat rusty and neither Johnstone nor the book in which this quote is embedded in expand upon 'the wonderful consequences'. What might they be?

It seems to me that one obvious result that would be a triviality is Baires Category theorem. Also, given that position and momentum observables are represented by densely defined unbounded operators, is this result useful there, directly or indirectly?

Second variation in saddle Finsler surface

Sun, 11/04/2018 - 01:57

Setting : Consider a two dimensional surface in $ (\mathbb{R}^n,\|\ \|)$. Here we define a function $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ s.t. $L(v)(X)=\langle f(v),X\rangle$ where $\langle\ ,\ \rangle$ is an inner product and $$L(v)(X)= \frac{d}{dt}\bigg|_{t=0}\ \frac{1}{2}\|v+tX\|^2 $$ Here note that $f(cv)=cf(v)$ for $c>0$.

Exercise : If $c$ is a path of unit speed from $p$ to $q$ in $M$, then $l(s)=\int_0^L \ \| c(t,s)_t\| \ dt$ where $c(t,s)$ is a variation of $c$. Then find a critical of a function $l$, called a geodesic

Proof : $l'(0)=\int\ L(c_t)(c_{ts})\ dt$. Here $\langle f(c_t),c_{ts}\rangle = \frac{d}{dt}\langle f(c_t),c_s\rangle - \langle df_{c_t} c_{tt},c_s\rangle$ so that $\langle df_{c_t} c_{tt},X\rangle=0$ for all $X\in T_{c(t,s)}M$ iff $c$ is a geodesic.

Remark : If $Q=df_{c_t}$, define $Q$-unit $n$ in direction $c_{tt}$, i.e. it is orthogonal to tangent space. If $ M$ is saddle, then $ \langle Qn,c_{st} \rangle^2 - \langle Q n,c_{tt} \rangle \langle Qn, c_{ss}\rangle >0$.

Question : Assume that $M$ is a saddle in $(\mathbb{R}^n,\|\ \|)$. Prove that any geodesic is minimizing.

Proof : Assume that $c(t,0)$ is a geodesic and is not minimizing between $c(0,0)$ and $c(L,0)$. If $l(s)=\int_0^L\ \|c(t,s)_t\|\ dt$, then $l''(0) <0$.

Reference : On intrinsic geometry of surface in normed spaces - Burago and Ivanv

Constant in the Marcinkiewicz-Zygmund inequality

Sat, 11/03/2018 - 23:10

Let $f : \mathbb{R} / \mathbb{Z} \to \mathbb{C}$ be a trigonometric polynomial of degree $n$ and $m-1 \geq n$ be an integer. The Marcinkiewicz-Zygmund inequality asserts $$\int |f|^p \leq \frac{C_p}{m} \sum_{j=1}^m |f(j/m)|^p , \ \ \ 1 < p < \infty.$$

My question is known about the behavior $C_p$, as a function of $p$? I am particularly interested in the case $1 < p < 2$.

Support of a regular measure Reg

Sat, 11/03/2018 - 20:39

Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of $K$ not necessarily in $\mathfrak{M}$. Suppose for all Baire sets $E\subseteq K\setminus S$, $\mu(E)=0$. Can I conclude that $Supp(\mu)\subset S$?

Problem in understanding means of i.i.d. variable

Sat, 11/03/2018 - 20:27

Let $X_i$ be i.i.d. and $X_i(\omega)= \omega, \omega \in (0,1)$, $P :=$ Lebesgue measure.

Then I have $\bar X_n (\omega) = \frac{\sum^n_{i=1} X_i(\omega)}{n} = \frac{n \omega }{n} = \omega$.

However, according to the law of large number, $\bar X_n \to EX$ in probability or almost surely under different condition. How is it possible??

In my example, $\bar X_n(\omega) \to \omega$. It is apparently not $EX$, which is a number!

Inverse Problem for Iwasawa Modules

Sat, 11/03/2018 - 20:26

Let $\Lambda$ denote the Iwasawa algebra and $M$ a finitely generated torsion $\Lambda$ module. Does there exist a number field $K$ and a $\mathbb{Z}_p$-extension $K_{\infty}/K$ such that the $p$-Hilbert class field $\Lambda$-module $X_{\infty}$ is pseudo-isomorphic to $M$? What about the same question for $\mathbb{Z}_p^2$-extensions?

Karp hardness of two cycles which lengths differ by one

Sat, 11/03/2018 - 20:15

Our problem is as follows:


Input: An undirected graph $G(V,E)$

Output: YES if there exists $2$ (simple) cycles in $G$ which lengths differ by $1$, otherwise NO

Is it $NP$-complete?

A bag contains 30 marbles [on hold]

Sat, 11/03/2018 - 19:57

15 are blue 10 are red 5 are black

How many permutations are there to choose six marbles where the first one is red?

how to find the system from observe data

Sat, 11/03/2018 - 18:39

original system contain two eigenvectors for each row but we can only observe one of them,

how to find the system which generate these two eigenvectors

Which partitions realise group algebras of finite groups?

Sat, 11/03/2018 - 16:22

Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the semisimple algebra $A_p:=M_{a_1}(K) \times \cdots \times M_{a_m}(K)$.


  1. Given a partition $p$, which finite groups $G$ have their group algebra over $K$ being isomorphic to $A_p$?

  2. Given a natural number $n$, how many partitions p with sum $n$ are there such that $A_p$ is isomorphic to a group algebra?

  3. How does the sequence of numbers of such partitions begin depending on $K$ (Does it in general depend on the field or perhaps just the characteristic of the field?)?

    Probably the answer is very complicated, but can something be said about the rough behavior of the sequence?

It should start as follows for $n \geq 1$ and $K=\mathbb{C}$, using GAP: 1,1,1,2,1,4,1,4,2,4,1,9,1,5,4

  1. $\textbf{Are those numbers always powers of primes?}$. (The next term takes forever to calculate, but maybe I should wait longer before asking this question....)

I dont know how complicated those questions are, so partial answers are also welcome.

Convexity of curves in Riemannian surfaces

Sat, 11/03/2018 - 15:29

It is known that a curve $f:[0,2\pi]\to \mathbf{R}^2$ is convex if $\partial_t (\arg f'(t))\ge 0$. My question is: does this statement have an analogue in the setting of Riemannian surfaces instead of $\mathbf{R}^2$?

Is it necessary for a conservative vector field to be $C^1$?

Sat, 11/03/2018 - 14:30

Is it necessary for a conservative vector field on a domain $A \subset \mathbb{R}^3$ to be $C^1$?

I know that if a vector field $F$ is defined on a simply connected domain $A$ simply connected and $F \in C^1(A)$, then $\operatorname{curl} F=0$ iff $F$ is conservative.

Is there a vector field F on a domain $A$ (I'm interested in both cases, $A$ simply connected or not) which is conservative but not $C^1$?

(Thanks, and sorry for my english)

Conjecture that relates matrix systems with some specific functions as solution sets

Sat, 11/03/2018 - 13:24

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in agreement with the conjecture . The conjecture is as follows : assume $x$ is a positive real variable that does not equal $1$ , and assume $y$ and $z$ are non-zero real variables , and consider for all $i, j \in \mathbb N$, $$a(i,j) = \frac{(x^{yi+z} + 1)^{j-1} + (x^y-1)}{x^y}$$ ; then for all $n \in \mathbb N$ , the solution set of the matrix system $[a(i,j) \mid 1 \leq i \leq n, 1 \leq j \leq (1+n)]$ exists and is unique with respect to $n$ and $x$ and $y$ and $z$ ,and each element in it is a sum of powers of $x$ with integer coefficients , and each of these powers of $x$ has the power as a linear combination of $y$ and $z$ such that the coefficients of $y$ and $z$ are non-negative integers .


For $n$=$7$, the solution set will be :

$s1$=x^yx^z + x^zx^(2*y) + x^z*x^(3*y) + x^z*x^(4*y) + x^z*x^(5*y) + x^z*x^(6*y) + x^z*x^(7*y) + x^(3*y)*x^(2*z) + x^(4*y)*x^(2*z) + 2*x^(5*y)*x^(2*z) + 2*x^(6*y)*x^(2*z) + x^(6*y)*x^(3*z) + 3*x^(7*y)*x^(2*z) + x^(7*y)*x^(3*z) + 3*x^(8*y)*x^(2*z) + 2*x^(8*y)*x^(3*z) + 3*x^(9*y)*x^(2*z) + 3*x^(9*y)*x^(3*z) + 2*x^(10*y)*x^(2*z) + 4*x^(10*y)*x^(3*z) + 2*x^(11*y)*x^(2*z) + x^(10*y)*x^(4*z) + 4*x^(11*y)*x^(3*z) + x^(12*y)*x^(2*z) + x^(11*y)*x^(4*z) + 5*x^(12*y)*x^(3*z) + x^(13*y)*x^(2*z) + 2*x^(12*y)*x^(4*z) + 4*x^(13*y)*x^(3*z) + 3*x^(13*y)*x^(4*z) + 4*x^(14*y)*x^(3*z) + 4*x^(14*y)*x^(4*z) + 3*x^(15*y)*x^(3*z) + 4*x^(15*y)*x^(4*z) + 2*x^(16*y)*x^(3*z) + x^(15*y)*x^(5*z) + 5*x^(16*y)*x^(4*z) + x^(17*y)*x^(3*z) + x^(16*y)*x^(5*z) + 4*x^(17*y)*x^(4*z) + x^(18*y)*x^(3*z) + 2*x^(17*y)*x^(5*z) + 4*x^(18*y)*x^(4*z) + 2*x^(18*y)*x^(5*z) + 3*x^(19*y)*x^(4*z) + 3*x^(19*y)*x^(5*z) + 2*x^(20*y)*x^(4*z) + 3*x^(20*y)*x^(5*z) + x^(21*y)*x^(4*z) + 3*x^(21*y)*x^(5*z) + x^(22*y)*x^(4*z) + x^(21*y)*x^(6*z) + 2*x^(22*y)*x^(5*z) + x^(22*y)*x^(6*z) + 2*x^(23*y)*x^(5*z) + x^(23*y)*x^(6*z) + x^(24*y)*x^(5*z) + x^(24*y)*x^(6*z) + x^(25*y)*x^(5*z) + x^(25*y)*x^(6*z) + x^(26*y)*x^(6*z) + x^(27*y)*x^(6*z) + x^(27*y)*x^(7*z) + 1

$s2$=- 6*x^y*x^z - 6*x^z*x^(2*y) - 6*x^z*x^(3*y) - 6*x^z*x^(4*y) - 6*x^z*x^(5*y) - 6*x^z*x^(6*y) - 6*x^z*x^(7*y) - 5*x^(3*y)*x^(2*z) - 5*x^(4*y)*x^(2*z) - 10*x^(5*y)*x^(2*z) - 10*x^(6*y)*x^(2*z) - 4*x^(6*y)*x^(3*z) - 15*x^(7*y)*x^(2*z) - 4*x^(7*y)*x^(3*z) - 15*x^(8*y)*x^(2*z) - 8*x^(8*y)*x^(3*z) - 15*x^(9*y)*x^(2*z) - 12*x^(9*y)*x^(3*z) - 10*x^(10*y)*x^(2*z) - 16*x^(10*y)*x^(3*z) - 10*x^(11*y)*x^(2*z) - 3*x^(10*y)*x^(4*z) - 16*x^(11*y)*x^(3*z) - 5*x^(12*y)*x^(2*z) - 3*x^(11*y)*x^(4*z) - 20*x^(12*y)*x^(3*z) - 5*x^(13*y)*x^(2*z) - 6*x^(12*y)*x^(4*z) - 16*x^(13*y)*x^(3*z) - 9*x^(13*y)*x^(4*z) - 16*x^(14*y)*x^(3*z) - 12*x^(14*y)*x^(4*z) - 12*x^(15*y)*x^(3*z) - 12*x^(15*y)*x^(4*z) - 8*x^(16*y)*x^(3*z) - 2*x^(15*y)*x^(5*z) - 15*x^(16*y)*x^(4*z) - 4*x^(17*y)*x^(3*z) - 2*x^(16*y)*x^(5*z) - 12*x^(17*y)*x^(4*z) - 4*x^(18*y)*x^(3*z) - 4*x^(17*y)*x^(5*z) - 12*x^(18*y)*x^(4*z) - 4*x^(18*y)*x^(5*z) - 9*x^(19*y)*x^(4*z) - 6*x^(19*y)*x^(5*z) - 6*x^(20*y)*x^(4*z) - 6*x^(20*y)*x^(5*z) - 3*x^(21*y)*x^(4*z) - 6*x^(21*y)*x^(5*z) - 3*x^(22*y)*x^(4*z) - x^(21*y)*x^(6*z) - 4*x^(22*y)*x^(5*z) - x^(22*y)*x^(6*z) - 4*x^(23*y)*x^(5*z) - x^(23*y)*x^(6*z) - 2*x^(24*y)*x^(5*z) - x^(24*y)*x^(6*z) - 2*x^(25*y)*x^(5*z) - x^(25*y)*x^(6*z) - x^(26*y)*x^(6*z) - x^(27*y)*x^(6*z) - 7

$s3$=15*x^y*x^z + 15*x^z*x^(2*y) + 15*x^z*x^(3*y) + 15*x^z*x^(4*y) + 15*x^z*x^(5*y) + 15*x^z*x^(6*y) + 15*x^z*x^(7*y) + 10*x^(3*y)*x^(2*z) + 10*x^(4*y)*x^(2*z) + 20*x^(5*y)*x^(2*z) + 20*x^(6*y)*x^(2*z) + 6*x^(6*y)*x^(3*z) + 30*x^(7*y)*x^(2*z) + 6*x^(7*y)*x^(3*z) + 30*x^(8*y)*x^(2*z) + 12*x^(8*y)*x^(3*z) + 30*x^(9*y)*x^(2*z) + 18*x^(9*y)*x^(3*z) + 20*x^(10*y)*x^(2*z) + 24*x^(10*y)*x^(3*z) + 20*x^(11*y)*x^(2*z) + 3*x^(10*y)*x^(4*z) + 24*x^(11*y)*x^(3*z) + 10*x^(12*y)*x^(2*z) + 3*x^(11*y)*x^(4*z) + 30*x^(12*y)*x^(3*z) + 10*x^(13*y)*x^(2*z) + 6*x^(12*y)*x^(4*z) + 24*x^(13*y)*x^(3*z) + 9*x^(13*y)*x^(4*z) + 24*x^(14*y)*x^(3*z) + 12*x^(14*y)*x^(4*z) + 18*x^(15*y)*x^(3*z) + 12*x^(15*y)*x^(4*z) + 12*x^(16*y)*x^(3*z) + x^(15*y)*x^(5*z) + 15*x^(16*y)*x^(4*z) + 6*x^(17*y)*x^(3*z) + x^(16*y)*x^(5*z) + 12*x^(17*y)*x^(4*z) + 6*x^(18*y)*x^(3*z) + 2*x^(17*y)*x^(5*z) + 12*x^(18*y)*x^(4*z) + 2*x^(18*y)*x^(5*z) + 9*x^(19*y)*x^(4*z) + 3*x^(19*y)*x^(5*z) + 6*x^(20*y)*x^(4*z) + 3*x^(20*y)*x^(5*z) + 3*x^(21*y)*x^(4*z) + 3*x^(21*y)*x^(5*z) + 3*x^(22*y)*x^(4*z) + 2*x^(22*y)*x^(5*z) + 2*x^(23*y)*x^(5*z) + x^(24*y)*x^(5*z) + x^(25*y)*x^(5*z) + 21

$s4$=- 20*x^y*x^z - 20*x^z*x^(2*y) - 20*x^z*x^(3*y) - 20*x^z*x^(4*y) - 20*x^z*x^(5*y) - 20*x^z*x^(6*y) - 20*x^z*x^(7*y) - 10*x^(3*y)*x^(2*z) - 10*x^(4*y)*x^(2*z) - 20*x^(5*y)*x^(2*z) - 20*x^(6*y)*x^(2*z) - 4*x^(6*y)*x^(3*z) - 30*x^(7*y)*x^(2*z) - 4*x^(7*y)*x^(3*z) - 30*x^(8*y)*x^(2*z) - 8*x^(8*y)*x^(3*z) - 30*x^(9*y)*x^(2*z) - 12*x^(9*y)*x^(3*z) - 20*x^(10*y)*x^(2*z) - 16*x^(10*y)*x^(3*z) - 20*x^(11*y)*x^(2*z) - x^(10*y)*x^(4*z) - 16*x^(11*y)*x^(3*z) - 10*x^(12*y)*x^(2*z) - x^(11*y)*x^(4*z) - 20*x^(12*y)*x^(3*z) - 10*x^(13*y)*x^(2*z) - 2*x^(12*y)*x^(4*z) - 16*x^(13*y)*x^(3*z) - 3*x^(13*y)*x^(4*z) - 16*x^(14*y)*x^(3*z) - 4*x^(14*y)*x^(4*z) - 12*x^(15*y)*x^(3*z) - 4*x^(15*y)*x^(4*z) - 8*x^(16*y)*x^(3*z) - 5*x^(16*y)*x^(4*z) - 4*x^(17*y)*x^(3*z) - 4*x^(17*y)*x^(4*z) - 4*x^(18*y)*x^(3*z) - 4*x^(18*y)*x^(4*z) - 3*x^(19*y)*x^(4*z) - 2*x^(20*y)*x^(4*z) - x^(21*y)*x^(4*z) - x^(22*y)*x^(4*z) - 35

$s5$=15*x^y*x^z + 15*x^z*x^(2*y) + 15*x^z*x^(3*y) + 15*x^z*x^(4*y) + 15*x^z*x^(5*y) + 15*x^z*x^(6*y) + 15*x^z*x^(7*y) + 5*x^(3*y)*x^(2*z) + 5*x^(4*y)*x^(2*z) + 10*x^(5*y)*x^(2*z) + 10*x^(6*y)*x^(2*z) + x^(6*y)*x^(3*z) + 15*x^(7*y)*x^(2*z) + x^(7*y)*x^(3*z) + 15*x^(8*y)*x^(2*z) + 2*x^(8*y)*x^(3*z) + 15*x^(9*y)*x^(2*z) + 3*x^(9*y)*x^(3*z) + 10*x^(10*y)*x^(2*z) + 4*x^(10*y)*x^(3*z) + 10*x^(11*y)*x^(2*z) + 4*x^(11*y)*x^(3*z) + 5*x^(12*y)*x^(2*z) + 5*x^(12*y)*x^(3*z) + 5*x^(13*y)*x^(2*z) + 4*x^(13*y)*x^(3*z) + 4*x^(14*y)*x^(3*z) + 3*x^(15*y)*x^(3*z) + 2*x^(16*y)*x^(3*z) + x^(17*y)*x^(3*z) + x^(18*y)*x^(3*z) + 35

$s6$=- 6*x^y*x^z - 6*x^z*x^(2*y) - 6*x^z*x^(3*y) - 6*x^z*x^(4*y) - 6*x^z*x^(5*y) - 6*x^z*x^(6*y) - 6*x^z*x^(7*y) - x^(3*y)*x^(2*z) - x^(4*y)*x^(2*z) - 2*x^(5*y)*x^(2*z) - 2*x^(6*y)*x^(2*z) - 3*x^(7*y)*x^(2*z) - 3*x^(8*y)*x^(2*z) - 3*x^(9*y)*x^(2*z) - 2*x^(10*y)*x^(2*z) - 2*x^(11*y)*x^(2*z) - x^(12*y)*x^(2*z) - x^(13*y)*x^(2*z) - 21

$s7$=x^yx^z + x^zx^(2*y) + x^z*x^(3*y) + x^z*x^(4*y) + x^z*x^(5*y) + x^z*x^(6*y) + x^z*x^(7*y) + 7

Thank you .