Algebraic Geometry – Macaulay2: How is the remainder calculated when a polynomial is divided by a set of polynomials (in a particular order)?

I write Buchberger's criterion into a program in Macaulay2 to see if the set of polynomials I have form a Grobner basis for the ideal that produces them. However, I have not been able to find a method that provides me with the rest, for example, if there is a polynomial $ f $is shared by a lot of polynomials $ G = {g_1, g_2, …, g_t} $ in a certain order. Would anyone know if such a method exists and if so, what is its name?

Although that's not all, here is the part of the program in which I try to implement the Buchberger criterion:

n=0;
for i to #polynomials-2 do
    (
        for j from i+1 to #polynomials-1 do 
    (
Spoly := lcm(leadTerm(polynomials#i),leadTerm(polynomials#j))/leadTerm(polynomials#i)*polynomials#i-lcm(leadTerm(polynomials#i),leadTerm(polynomials#j))/leadTerm(polynomials#j)*polynomials#j;
remainder := Spair%polynomials;
if remainder == 0 then n=n+1;
  );
    );
    if n == binomial(#polynomials, 2) then print "The polynomials form a Grobner basiss for the ideal it generates." else print "The polynomials don't form a Grobner basis for the ideal it generates."

What is the reason why America is so divided?

Politics was ALWAYS splitting. Damn, some of our founders killed each other in DUELS. There were fist fights on the floor of the congress.

However, I think we are experiencing a time when both parties and their constituents are moving farther and farther from the center. I have three main reasons.

1) Computer Aided Partisan Gerry Mymandering. Partisan walking is nothing new. What's different is that computers and demographic data now allow consultants to draw hyper-accurate maps to achieve VERY specific election results. Parliamentary elections are therefore safer than ever. Incumbents, who ALWAYS lose, do so in their primaries. SCOTUS has recently dealt with it, and I can not question its legal logic. But that's a big problem and both parties are doing it.

Safe parliamentary elections and looming primaries mean that the average politician is threatened only by his extreme flank. You do not have to worry about a centrist threat. So they become more extreme to protect themselves from the primary challengers.

2) McCain fine gold and other campaign financing reform. Every time someone talks to you about taking money out of politics, you should slap him in the mouth. Any reform of campaign financing has made established operators safer, and the largest of these was McCain Fine Gold (MF).

MF's main contribution was to eliminate soft money, ie people who donated to parties rather than to candidates, and then the parties who paid the money. Unlike hard money (which went to the candidates), the campaign campaigners said it was a source of corruption. That was a lie.

Soft money was almost always spent by the party on behalf of CHALLENGERS, not office-holders. The established operators have an advantage in terms of money and fame and usually need no help. So, parties usually spend money on challengers. In other words, with soft money, elections have become more COMPETITIVE. Which is why established companies hate soft money.

3) Modern media. Let's face it, the media in America have never been as good as people sometimes think. Yellow journalism and the use of newspapers for personal attacks on vendettes are as old as the existence of the printing house. Maybe older.

But what has changed is that eventually confidence in journalists has grown. A misplaced belief. And the industry has since fragmented and decayed. Now everyone selects his news source to confirm his own prejudices. Liberals only listen to liberal biased sources, conservatives only read conservatively biased sources.

And across all sources the journalistic responsibility is gone. The days when each article needs two or more RELIABLE sources to print are over. Now even the slightest rumor goes straight into the air. Being FIRST is more important than being RIGHT. And retractions are rarely and never prominently displayed. It is not uncommon for the opinion to be a journalist, and too many viewers can not say where one ends and the other begins.

As people choose their own sources to counter their bias, most people live in echo chambers where they do not even hear hostile ideas in their original form. Instead, they listen to opposing ideas at best in a slanted derogatory form that ensures no one can find common ground.

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Why do liberals pretend that the country was not already divided when Trump took office?

No, you idiot, it was YOUR reaction to Obama, which has led to divisions. It was HER people who hung him up in a picture, HER people who made jokes about having a watermelon bed and a roast chicken on the White House lawn, HER people, him, his wife, and his daughters with monkeys compared. It was YOU who vowed to make him a president even before he took office. It was YOU who claimed he was not a citizen. You attacked him because you did exactly what you praised GW Bush for. You can not blame Obama for what YOU have done. You people are the crass, bigoted ignoramuses who have been practicing non-stop racial lures for 8 years. It was simply in your throng, a vicious black man who was elected president rather than "knowing his right place." GFY. You people are not unfortunate, you are scornful, you should be ashamed, but you are not. You are proud to be like that. You are a shame for the USA.

But the split goes back even further. It really started with Rush Limbaugh in the 80s, then Newt Gingrich in the 90s – this idea that Democrats were not just the loyal political opposition, but the "enemy". YOU @ssholes started it all, then you have the balls to try and blame us.

You are the one without a clue.

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Expectation of X divided by Y

Suppose we have two continuous independent random variables $ X $ and $ Y $, I understand that

$ E[XY] = E[X]e[Y]$

but what about $ E[X/Y]$, Can I say:

$ E[Xg(Y)]$ from where $ g (x) = dfrac {1} {x} $

$ E[X]e[g(Y)]$ through linearity

Then solve for everyone? If not, how do you go about solving this type of problem?

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