How to prove that the sequence is regular? This definition of regular sequence that I know is:
Let $R$ be a ring and let $M$ be an $R$-module. A sequence of
elements $\{x_1,... ,x_n\} \in R$ is called a regular sequence on $M$ (or an $M$- sequence) if

*

*$(x_1,..., x_n)M \ne M$, and

*For $i = 1,... ,n$, $x_i$ is a nonzerodivisor on $M/(x_1,... ,x_{i-1})M$.


My question is to show that in the ring $K[x,y,z]$ the sequence $x,x+y^2,x+y+z^3$ is regular.

This is not a homework problem. I am trying to self-study commutative algebra and this definition gives me no idea on how to even start this problem. Can someone show me how to do this problem with some details?
My original question asks me to show that
$\{y_1+x_1y_2+x_1^2y_3,y_1+x_2y_2+x_2^2y_3,y_1+x_3y_2+x_3^2y_3\}$ is a regular sequence in $k[y_1,y_2,y_3,x_1,x_2,x_3]$ (the order is lexico)
Going exactly in the way @Joshua suggested I did consider the quotient ring $k[y_1,y_2,y_3,x_1,x_2,x_3]/(y_1+x_1y_2+x_1^2y_3)$. The image of the second element that I got is $(x_2-x_1)(y_2+y_3x_2+y_3x_1)$ I am not sure how to proceed,  is there a computer code or something that he will help me?
The Grobner basis of $I=\left<f_1,f_2,f_3\right>$ that I got is:

*

*$y_1+ y_2x_3 + y_3x_3^2$


*$-y_2x_3 + y_2x_1 - y_3x_3^2 + y_3x_1^2$


*$-y_2x_3 + y_2x_2 - y_3x_3^2 + y_3x_2^2$


*$y_3x_1^2x_2 - y_3x_1^2x_3 - y_3x_1x_2^2 + y_3x_1x_3^2 + y_3x_2^2x_3 - y_3x_2x_3^2$
The monomial order is $y_1 > y_2 > y_3 > x_1 > x_2 > x_3$
What is a Hilbert series? How do you compute it with the help of Grobner techniques? Do you have any access to any example which can help me with this (maybe any worked out example)? I went through Stanley's paper but I think that will need a strong background in commutative algebra. (I only know upto Grobner basis which I read from David A. Cox.)
 A: In this case you can directly apply the definition.
$K[x, y, z]/(x) = K[y, z]$, of course, and the image of $x+y^2$ is just $y^2$, which is clearly not a zero-divisor. Next, $K[y, z]/(y^2)$ has $K$-basis $\{1, z, z^2, \ldots, y, yz, yz^2, \ldots\}$ subject to the relation $y^2 = 0$ and the image of $x+y+z^3$ is $y+z^3$. Suppose $y+z^3$ were a zero-divisor. Say $(f(z) + yg(z))(y+z^3) = 0$. Then $f(z)z^3 + y(f(z) + g(z)z^3) = 0$. That forces $f(z)z^3 = 0$ and $f(z) + g(z)z^3 = 0$. But that forces $f(z)=0$ and then $g(z)=0$. So $y+z^3$ is not a zerodivisor.
Edit in response to follow-up question in comments: Sequences of homogeneous elements are better-behaved with respect to being regular, e.g. they're regular if and only if any rearrangement is. In your case you could declare $\deg y_1 = 3, \deg y_2 = 2, \deg y_3 = \deg x_i = 1$, making your elements $f_1, f_2, f_3$ all homogeneous of degree $3$. Now you can compute the Hilbert series of $K[x_1, x_2, x_3, y_1, y_2, y_3]/(f_1, f_2, f_3)$ using Grobner-theoretic techniques and check if it's correct*. This is apparently what Macaulay2 does.
That could technically be done by hand simply using Buchberger's algorithm and manual S-polynomial reductions (I'm guessing the Grobner basis here will be quite small), but it's very tedious and error-prone. Once you get the Grobner basis, then you'd just need to pick off their leading terms, give an explicit description of the standard monomials (those that aren't divisible by any of those leading terms), and compute the Hilbert series.
I wouldn't be surprised if mucking about along the lines of my original answer were still faster in this particular case.
*What is the correct Hilbert series? See e.g. Corollary 3.2 of Stanley's lovely mostly-survey paper "Hilbert Functions of Graded Algebras".
