Polyphony Digital has changed the game after apologizing in March.

To make amends for some disastrous downtime, Gran Turismo 7 director Kazunori Yamauchi announced a credits handout and promised to “dramatically change GT7's car economy to help make amends” last month. The first of these has arrived.

The game's 1.11 update includes the following concessions to players frustrated by the economy and its subsequent grind:

• The last half of the World Circuits events have increased in-game credit rewards.

• Modified Arcade and Custom Race rewards

• Clearing all circuit layouts with Gold or Bronze now rewards In-game Credits. Exiting the Sector selection screen with the Exit button will award Credits if an event has already been cleared.

• Increased Credits Rewards in Lobby and Daily Races

• Increased the free in-game Credits cap from 20,000,000 to 100,000,000.

Additionally, “The Human Comedy” missions are one-hour endurance races that award “up to 1,200,000” credits per event.

This isn't everything Yamauchi promised last month; he said it would take several patches and updates to fully implement the changes. Here's a list of everything he said would happen, some of which have already happened (like the World Cup rewards and credit cap):

• Increase rewards in the latter half of the World Circuits by roughly 100%.
• Added high rewards for all Gold/Bronze results clearing the Circuit Experience.
• Online Races rewards increase.
• Add 8 new 1-hour Endurance Race events to Missions. So expect higher rewards.
• Increase the non-paid credit limit in player wallets from 20M to 100M.
• Expand the number of Used and Legend cars available at any time.
• With time, we will increase the payout value of limited time rewards.
• New World Circuit events.
• Missions now include 24-hour endurance races.
• Online Time Trials added, with rewards based on the player's time difference from the leader.
• Make cars sellable.

The full list of updates and changes can be found here.

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If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

• x+5
• x^4
• x^3+3x^2+3x+1
• 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

• A(0)+B(0)=C(0)
• A(1)+B(1)=C(1)
• A(2)+B(2)=C(2)
• A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

• 12+1=13
• 10+8=18
• 15+8=23
• 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

• A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
• B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
• C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

Since the pandemic, I've worked from home. It’s been +2 years (wow, time flies!) now, and during this time I’ve learned a lot. My 4 remote work lessons.

I work in a remote distributed team. This team setting shaped my experience and teachings.

Isolation ("I miss my coworkers")

The most obvious point. I miss going out with my coworkers for coffee, weekend chats, or just company while I work. I miss being able to go to someone's desk and ask for help. On a remote world, I must organize a meeting, share my screen, and avoid talking over each other in Zoom - sigh!

Social interaction is more vital for my health than I believed.

Online socializing stinks

My company used to come together every Friday to play Exploding Kittens, have food and beer, and bond over non-work things.

Different today. Every Friday afternoon is for fun, but it's not the same. People with screen weariness miss meetings, which makes sense. Sometimes you're too busy on Slack to enjoy yourself.

We laugh in meetings, but it's not the same as face-to-face.

Digital social activities can't replace real-world ones

Improved Work-Life Balance, if You Let It

At the outset of the pandemic, I recognized I needed to take better care of myself to survive. After not leaving my apartment for a few days and feeling miserable, I decided to walk before work every day. This turned into a passion for exercise, and today I run or go to the gym before work. I use my commute time for healthful activities.

Working from home makes it easier to keep working after hours. I sometimes forget the time and find myself writing coding at dinnertime. I said, "One more test." This is a disadvantage, therefore I keep my office schedule.

Spend your commute time properly and keep to your office schedule.

Remote Pair Programming Is Hard

As a software developer, I regularly write code. My team sometimes uses pair programming to write code collaboratively. One person writes code while another watches, comments, and asks questions. I won't list them all here.

Internet pairing is difficult. My team struggles with this. Even with Tuple, it's challenging. I lose attention when I get a notification or check my computer.

I miss a pen and paper to rapidly sketch down my thoughts for a colleague or a whiteboard for spirited talks with others. Best answers are found through experience.

Real-life pair programming beats the best remote pair programming tools.

Lessons Learned

Here are 4 lessons I've learned working remotely for 2 years.

• Socializing is more vital to my health than I anticipated.

• Digital social activities can't replace in-person ones.

• Spend your commute time properly and keep your office schedule.

• Real-life pair programming beats the best remote tools.

Conclusion

Our era is fascinating. Remote labor has existed for years, but software companies have just recently had to adapt. Companies who don't offer remote work will lose talent, in my opinion.

We're still figuring out the finest software development approaches, programming language features, and communication methods since the 1960s. I can't wait to see what advancements assist us go into remote work.

I'll certainly work remotely in the next years, so I'm interested to see what I've learnt from this post then.

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